Difference between revisions of "Line Integrals"
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The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on intervals. Many simple formulae in physics, such as the definition of work as <math>W=\mathbf{F}\cdot\mathbf{s}</math>, have natural continuous analogues in terms of line integrals, in this case <math>\textstyle W = \int_L \mathbf{F}(\mathbf{s})\cdot d\mathbf{s}</math>, which computes the work done on an object moving through an electric or gravitational field '''F''' along a path <math>L</math>. | The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on intervals. Many simple formulae in physics, such as the definition of work as <math>W=\mathbf{F}\cdot\mathbf{s}</math>, have natural continuous analogues in terms of line integrals, in this case <math>\textstyle W = \int_L \mathbf{F}(\mathbf{s})\cdot d\mathbf{s}</math>, which computes the work done on an object moving through an electric or gravitational field '''F''' along a path <math>L</math>. | ||
| + | |||
| + | == Volume, path (line), and surface integrals == | ||
| + | |||
| + | === Volume Integrals === | ||
| + | |||
| + | Volume integrals have already been discussed in the chapter [[Calculus/Multivariable calculus|Multivariable calculus]], but a brief review is given here for completeness. | ||
| + | |||
| + | Given a scalar field <math>\rho: \R^3 \to \R</math> that denotes a density at each specific point, and an arbitrary volume <math>\Omega \subseteq \R^3</math>, the total "mass" <math>M</math> inside of <math>\Omega</math> can be determined by partitioning <math>\Omega</math> into infinitesimal volumes. At each position <math>\mathbf{q} \in \Omega</math>, the volume of the infinitesimal volume is denoted by the infinitesimal <math>dV</math>. This gives rise to the following integral: | ||
| + | |||
| + | <math> M = \iiint_{\mathbf{q} \in \Omega} \rho(\mathbf{q})dV </math> | ||
| + | |||
| + | === Path Integrals === | ||
| + | |||
| + | Given any oriented path <math>C</math> (oriented means that there is a preferred direction), the differential <math>d\mathbf{q} = dx\mathbf{i} + dy\mathbf{j} + dz\mathbf{k}</math> denotes an infinitesimal displacement along <math>C</math> in the preferred direction. This differential can be used in various path integrals. Letting <math>f: \R^3 \to \R</math> denote an arbitrary scalar field, and <math>\mathbf{F}: \R^3 \to \R^3</math> denote an arbitrary vector field, various path integrals include: | ||
| + | |||
| + | <math>\int_{\mathbf{q} \in C} f(\mathbf{q})d\mathbf{q}</math>, <math>\int_{\mathbf{q} \in C} f(\mathbf{q})|d\mathbf{q}|</math>, <math>\int_{\mathbf{q} \in C} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q}</math>, <math>\int_{\mathbf{q} \in C} \mathbf{F}(\mathbf{q})|d\mathbf{q}|</math>, and many more. | ||
| + | |||
| + | <math>\int_{\mathbf{q} \in C} d\mathbf{q}</math> denotes the total displacement along <math>C</math>, and <math>\int_{\mathbf{q} \in C} |d\mathbf{q}|</math> denotes the total length of <math>C</math>. | ||
| + | |||
| + | ==== Calculating Path Integrals ==== | ||
| + | |||
| + | To compute a path integral, the continuous oriented curve <math>C</math> must be parameterized. <math>\mathbf{q}_C(t)</math> will denote the point along <math>C</math> indexed by <math>t</math> from the range <math>[t_0, t_1]</math>. <math>\mathbf{q}_C(t_0) = \mathbf{q}_0</math> must be the starting point of <math>C</math> and <math>\mathbf{q}_C(t_1) = \mathbf{q}_1</math> must be the ending point of <math>C</math>. As <math>t</math> increases, <math>\mathbf{q}_C(t)</math> must proceed along <math>C</math> in the preferred direction. An infinitesimal change in <math>t</math>, <math>dt</math>, results in the infinitesimal displacement <math>d\mathbf{q} = \frac{d\mathbf{q}_C}{dt}dt</math> along <math>C</math>. In the path integral <math>\int_{\mathbf{q} \in C} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q}</math>, the differential <math>d\mathbf{q}</math> can be replaced with <math>\frac{d\mathbf{q}_C}{dt}dt</math> to get <math>\int_{t = t_0}^{t_1} \mathbf{F}(\mathbf{q}_C(t)) \cdot \frac{d\mathbf{q}_C}{dt}dt</math> | ||
| + | |||
| + | {{DropBox|Example 1| | ||
| + | As an example, consider the vector field <math>\mathbf{F}(x,y,z) = 3\mathbf{i} - x\mathbf{j} + 5y\mathbf{k}</math> and the straight line curve <math>C</math> that starts at <math>(1,1,1)</math> and ends at <math>(7,-1,-2)</math>. <math>C</math> can be parameterized by <math>\mathbf{q}_C(t) = (1+6t, 1-2t, 1-3t)</math> where <math>t \in [0, 1]</math>. | ||
| + | <math>\frac{d\mathbf{q}_C}{dt} = 6\mathbf{i} - 2\mathbf{j} - 3\mathbf{k}</math>. | ||
| + | We can then evaluate the path integral <math>\int_{\mathbf{q} \in C} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q}</math> as follows: | ||
| + | |||
| + | <math>\int_{\mathbf{q} \in C} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q} | ||
| + | = \int_{\mathbf{q} \in C} (3\mathbf{i} - x\mathbf{j} + 5y\mathbf{k}) \cdot d\mathbf{q} | ||
| + | = \int_{t=0}^{1} (3\mathbf{i} - (1+6t)\mathbf{j} + 5(1-2t)\mathbf{k}) \cdot (6\mathbf{i} - 2\mathbf{j} - 3\mathbf{k})dt</math> | ||
| + | |||
| + | <math> = \int_{t=0}^{1} (18 + (2+12t) + (-15+30t))dt | ||
| + | = \int_{t=0}^{1} (5 + 42t)dt | ||
| + | = (5t + 21t^2)\bigg|_{t=0}^1 | ||
| + | = 26</math> | ||
| + | |hidden=hidden}} | ||
| + | |||
| + | If a vector field <math>\mathbf{F}</math> denotes a "force field", which returns the force on an object as a function of position, the [[w:Work_(physics)|work]] performed on a point mass that traverses the oriented curve <math>C</math> is <math>W = \int_{\mathbf{q} \in C} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q}</math> | ||
| + | |||
| + | {{DropBox|Example 2| | ||
| + | Consider the gravitational field that surrounds a point mass of <math>M</math> located at the origin: <math>\mathbf{g}(\mathbf{q}) = -\frac{GM}{|\mathbf{q}|^2}\frac{\mathbf{q}}{|\mathbf{q}|}</math> using [[w:Newton%27s_law_of_universal_gravitation|Newton's inverse square law]]. The force acting on a point mass of <math>m</math> at position <math>\mathbf{q}</math> is <math>\mathbf{F}(\mathbf{q}) = m\mathbf{g}(\mathbf{q}) = -\frac{GMm}{|\mathbf{q}|^2}\frac{\mathbf{q}}{|\mathbf{q}|}</math>. In spherical coordinates the force is <math>\mathbf{F}(r,\theta,\phi) = -\frac{GMm}{r^2}\hat{\mathbf{r}}</math> (note that <math>\hat{\mathbf{r}}, | ||
| + | \hat{\mathbf{\theta}}, \hat{\mathbf{\phi}}</math> are the unit length mutually orthogonal basis vectors for spherical coordinates). | ||
| + | |||
| + | Consider an arbitrary path <math>C</math> that <math>m</math> traverses that starts at an altitude of <math>r = r_1</math> and ends at an altitude of <math>r = r_2</math>. The work done by the gravitational field is: | ||
| + | |||
| + | <math>W = \int_{\mathbf{q} \in C} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q}</math> | ||
| + | |||
| + | The infinitesimal displacement <math>d\mathbf{q}</math> is equivalent to the displacement expressed in spherical coordinates: <math>dr \cdot \hat{\mathbf{r}} + r \cdot d\theta \cdot \hat{\mathbf{\theta}} + r\sin\theta \cdot d\phi \cdot \hat{\mathbf{\phi}}</math>. | ||
| + | |||
| + | <math>W = \int_{\mathbf{q} \in C} -\frac{GMm}{r^2}\hat{\mathbf{r}} \cdot (dr \cdot \hat{\mathbf{r}} + r \cdot d\theta \cdot \hat{\mathbf{\theta}} + r\sin\theta \cdot d\phi \cdot \hat{\mathbf{\phi}}) = \int_{r=r_1}^{r_2} -\frac{GMm}{r^2}dr = \frac{GMm}{r}\bigg|_{r=r_1}^{r_2} = GMm\left(\frac{1}{r_2} - \frac{1}{r_1}\right)</math> | ||
| + | |||
| + | The work is equal to the amount of gravitational potential energy lost, so one possible function for the gravitational potential energy is <math>\phi(r,\theta,\phi) = -\frac{GMm}{r}</math> or equivalently, <math>\phi(\mathbf{q}) = -\frac{GMm}{|\mathbf{q}|}</math>. | ||
| + | |hidden=hidden}} | ||
| + | |||
| + | |||
| + | {{DropBox|Example 3| | ||
| + | Consider the spiral <math>C</math> parameterized with respect to <math>t</math> in cylindrical coordinates by <math>\mathbf{q}_C(t) = (\rho = R, \phi = t, z = k\frac{t}{2\pi})</math>. Consider the problem of determining the spiral's length with <math>t</math> restricted to the range <math>[0, 2\pi]</math>. An infinitesimal change of <math>dt</math> in <math>t</math> results in the infinitesimal displacement: | ||
| + | |||
| + | <math>d\mathbf{q} = \frac{d\mathbf{q}_C}{dt}dt = \left(\frac{d\rho}{dt}\hat{\mathbf{\rho}} + \rho\frac{d\phi}{dt}\hat{\mathbf{\phi}} + \frac{dz}{dt}\hat{\mathbf{z}}\right)dt = \left(R\hat{\mathbf{\phi}} + \frac{k}{2\pi}\hat{\mathbf{z}}\right)dt</math> | ||
| + | |||
| + | The length of the infinitesimal displacement is <math>|d\mathbf{q}| = \sqrt{R^2 + \left(\frac{k}{2\pi}\right)^2} \cdot dt</math>. | ||
| + | |||
| + | The length of the spiral is therefore: <math>\int_{\mathbf{q} \in C} |d\mathbf{q}| = \int_{t=0}^{2\pi} \sqrt{R^2 + \left(\frac{k}{2\pi}\right)^2} \cdot dt = 2\pi\sqrt{R^2 + \left(\frac{k}{2\pi}\right)^2} = \sqrt{(2\pi R)^2 + k^2}</math> | ||
| + | |||
| + | |hidden=hidden}} | ||
| + | |||
| + | === Surface Integrals === | ||
| + | |||
| + | Given any oriented surface <math>\sigma</math> (oriented means that the there is a preferred direction to pass through the surface), an infinitesimal portion of the surface is defined by an infinitesimal area <math>|dS|</math>, and a unit length outwards oriented normal vector <math>\mathbf{n}</math>. <math>\mathbf{n}</math> has a length of 1 and is perpendicular to the surface of <math>\sigma</math>, while penetrating <math>\sigma</math> in the preferred direction. The infinitesimal portion of the surface is denoted by the infinitesimal "surface vector": <math>\mathbf{dS} = |dS|\mathbf{n}</math>. If a vector field <math>\mathbf{F}: \R^3 \to \R^3</math> denotes a flow density, then the flow through the infinitesimal surface portion in the preferred direction is <math>\mathbf{F}(\mathbf{q}) \cdot \mathbf{dS}</math>. | ||
| + | |||
| + | The infinitesimal "surface vector" <math>\mathbf{dS} = \mathbf{n}|dS|</math> describes the infinitesimal surface element in a manner similar to how the infinitesimal displacement <math>d\mathbf{q}</math> describes an infinitesimal portion of a path. More specifically, similar to how the interior points on a path do not affect the total displacement, the interior points on a surface to not affect the total surface vector. | ||
| + | |||
| + | [[File:Equivalence of total displacement.svg|thumb|The displacement between two points is independent of the path that connects them.]] | ||
| + | |||
| + | Consider for instance two paths <math>C_1</math> and <math>C_2</math> that both start at point <math>A</math>, and end at point <math>B</math>. The total displacements, <math>\int_{\mathbf{q} \in C_1} d\mathbf{q}</math> and <math>\int_{\mathbf{q} \in C_2} d\mathbf{q}</math>, are both equivalent and equal to the displacement between <math>A</math> and <math>B</math>. Note however that the total lengths <math>\int_{\mathbf{q} \in C_1} |d\mathbf{q}|</math> and <math>\int_{\mathbf{q} \in C_2} |d\mathbf{q}|</math> are not necessarily equivalent. | ||
| + | |||
| + | Similarly, given two surfaces <math>S_1</math> and <math>S_2</math> that both share the same counter-clockwise oriented boundary <math>C</math>, the total surface vectors <math>\int_{\mathbf{q} \in S_1} \mathbf{dS}</math> and <math>\int_{\mathbf{q} \in S_2} \mathbf{dS}</math> are both equivalent and are a function of the boundary <math>C</math>. This implies that a surface can be freely deformed within its boundaries without changing the total surface vector. Note however that the surface areas <math>\int_{\mathbf{q} \in S_1} |\mathbf{dS}|</math> and <math>\int_{\mathbf{q} \in S_2} |\mathbf{dS}|</math> are not necessarily equivalent. | ||
| + | |||
| + | The fact that the total surface vectors of <math>S_1</math> and <math>S_2</math> are equivalent is not immediately obvious. To prove this fact, let <math>\mathbf{F}</math> be a constant vector field. <math>S_1</math> and <math>S_2</math> share the same boundary, so the flux/flow of <math>\mathbf{F}</math> through <math>S_1</math> and <math>S_2</math> is equivalent. The flux through <math>S_1</math> is <math>\Phi_1 = \int_{\mathbf{q} \in S_1} \mathbf{F} \cdot \mathbf{dS} = \mathbf{F} \cdot \int_{\mathbf{q} \in S_1} \mathbf{dS} </math>, and similarly for <math>S_2</math> is <math>\Phi_2 = \int_{\mathbf{q} \in S_2} \mathbf{F} \cdot \mathbf{dS} = \mathbf{F} \cdot \int_{\mathbf{q} \in S_2} \mathbf{dS} </math>. Since <math> \mathbf{F} \cdot \int_{\mathbf{q} \in S_1} \mathbf{dS} = \mathbf{F} \cdot \int_{\mathbf{q} \in S_2} \mathbf{dS} </math> for every choice of <math>\mathbf{F}</math>, it follows that <math> \int_{\mathbf{q} \in S_1} \mathbf{dS} = \int_{\mathbf{q} \in S_2} \mathbf{dS} </math>. | ||
| + | |||
| + | The geometric significance of the total surface vector is that each component measures the area of the projection of the surface onto the plane formed by the other two dimensions. Let <math>\sigma</math> be a surface with surface vector <math>\mathbf{S} = S_x\mathbf{i} + S_y\mathbf{j} + S_z\mathbf{k}</math>. It is then the case that: <math>S_x</math> is the area of the projection of <math>\sigma</math> onto the yz-plane; <math>S_y</math> is the area of the projection of <math>\sigma</math> onto the xz-plane; and <math>S_z</math> is the area of the projection of <math>\sigma</math> onto the xy-plane. | ||
| + | |||
| + | [[Image:Stokes'_Theorem.svg|thumb|right|250px|The boundary <math>\partial\Sigma</math> of <math>\Sigma</math> is counter-clockwise oriented.]] | ||
| + | |||
| + | Given an oriented surface <math>\Sigma</math>, another important concept is the oriented boundary. The boundary of <math>\Sigma</math> is an oriented curve <math>\partial\Sigma</math> but how is the orientation chosen? If the boundary is "counter-clockwise" oriented, then the boundary must follow a counter-clockwise direction when the oriented surface normal vectors point towards the viewer. The counter-clockwise boundary also obeys the "right-hand rule": If you hold your right hand with your thumb in the direction of the surface normals (penetrating the surface in the "preferred" direction), then your fingers will wrap around in the direction of the counter-clockwise oriented boundary. | ||
| + | |||
| + | {{DropBox|Example 1| | ||
| + | Consider the Cartesian points <math>(0,0,0)</math>; <math>(1,0,0)</math>; <math>(0,1,0)</math>; and <math>(0,0,1)</math>. | ||
| + | |||
| + | Let <math>\sigma_1</math> be the surface formed by the triangular planes <math>\{(0,1,0), (0,0,0), (1,0,0)\}</math>; <math>\{(0,0,1), (0,0,0), (0,1,0)\}</math>; and <math>\{(1,0,0), (0,0,0), (0,0,1)\}</math> where the vertices are listed in a counterclockwise direction relative to the surface normal directions. The surface vectors of each plane are respectively <math>\frac{1}{2}\mathbf{k}</math>; <math>\frac{1}{2}\mathbf{i}</math>; and <math>\frac{1}{2}\mathbf{j}</math> respectively which add to a total surface vector of <math>\mathbf{S}_1 = \frac{1}{2}(\mathbf{i}+\mathbf{j}+\mathbf{k})</math>. | ||
| + | |||
| + | Let <math>\sigma_2</math> be the surface formed by the single triangular plane <math>\{(1,0,0), (0,1,0), (0,0,1)\}</math> where the vertices are listed in a counterclockwise direction relative to the normal direction. It can be seen that <math>\sigma_1</math> and <math>\sigma_2</math> share a the common counter clockwise boundary <math>(1,0,0) \to (0,1,0) \to (0,0,1)</math>The surface vector is <math>\mathbf{S}_2 = \frac{1}{2}(\mathbf{j} - \mathbf{i}) \times (\mathbf{k} - \mathbf{i}) = \frac{1}{2}(\mathbf{i} + \mathbf{j} + \mathbf{k})</math> which is equivalent to <math>\mathbf{S}_1</math>. | ||
| + | |hidden=hidden}} | ||
| + | |||
| + | {{DropBox|Example 2| | ||
| + | This example will show how moving a point that is in the interior of a "triangular mesh" does not affect the total surface vector. Consider the points <math>P_0, P_1, P_2, \dots, P_n</math> where <math>n \geq 3</math>. Let the closed path <math>C</math> be defined by the cycle <math>P_1 \to P_2 \to \dots \to P_n \to P_1</math>. For simplicity, <math>P_{n+1} = P_1</math>. For each <math>i = 1, 2, \dots, n</math>, <math>\mathbf{v}_i</math> will denote the displacement of <math>P_i</math> relative to <math>P_0</math>. Like with <math>P_{n+1}</math>, <math>\mathbf{v}_{n+1} = \mathbf{v}_1</math>. | ||
| + | |||
| + | Let <math>\sigma</math> denote a surface that is a "triangular mesh" comprised of the closed fan of triangles: <math>\{P_2,P_0,P_1\}</math>; <math>\{P_3,P_0,P_2\}</math>; ...; <math>\{P_n,P_0,P_{n-1}\}</math>; <math>\{P_1,P_0,P_n\}</math> where the vertices of each triangle are listed in a counterclockwise direction. It can be seen that the counterclockwise boundary of <math>\sigma</math> is <math>C</math> and does not depend on the location of <math>P_0</math>. The total surface vector for <math>\sigma</math> is: | ||
| + | |||
| + | <math>\mathbf{S} = \frac{1}{2}(\mathbf{v}_1 \times \mathbf{v}_2) + \frac{1}{2}(\mathbf{v}_2 \times \mathbf{v}_3) + \dots + \frac{1}{2}(\mathbf{v}_{n-1} \times \mathbf{v}_n) + \frac{1}{2}(\mathbf{v}_n \times \mathbf{v}_{n+1}) = \frac{1}{2}\sum_{i=1}^n (\mathbf{v}_i \times \mathbf{v}_{i+1})</math> | ||
| + | |||
| + | Now displace <math>P_0</math> by <math>\mathbf{w}</math> to get <math>P'_0</math>. The displacement vector of <math>P_i</math> relative to <math>P_0'</math> becomes <math>\mathbf{v}'_i = \mathbf{v}_i - \mathbf{w}</math>. The counterclockwise boundary is unaffected. The total surface vector is: | ||
| + | |||
| + | <math>\mathbf{S}' = \frac{1}{2}\sum_{i=1}^n (\mathbf{v}'_i \times \mathbf{v}'_{i+1}) | ||
| + | = \frac{1}{2}\sum_{i=1}^n ((\mathbf{v}_i - \mathbf{w}) \times (\mathbf{v}_{i+1} - \mathbf{w})) | ||
| + | = \frac{1}{2}\sum_{i=1}^n ((\mathbf{v}_i \times \mathbf{v}_{i+1}) - (\mathbf{v}_i \times \mathbf{w}) - (\mathbf{w} \times \mathbf{v}_{i+1}) + (\mathbf{w} \times \mathbf{w})) </math> | ||
| + | |||
| + | <math> = \frac{1}{2}\sum_{i=1}^n ((\mathbf{v}_i \times \mathbf{v}_{i+1}) - \mathbf{w} \times (\mathbf{v}_{i+1} - \mathbf{v}_i)) | ||
| + | = \mathbf{S} - \frac{1}{2}\mathbf{w} \times (\mathbf{v}_{n+1} - \mathbf{v}_1) = \mathbf{S} - \frac{1}{2}\mathbf{w} \times \mathbf{0} = \mathbf{S}</math> | ||
| + | |||
| + | Therefore moving the interior point <math>P_0</math> neither affects the boundary, nor the total surface vector. | ||
| + | |||
| + | |hidden=hidden}} | ||
| + | |||
| + | |||
| + | ==== Calculating Surface Integrals ==== | ||
| + | |||
| + | To calculate a surface integral, the oriented surface <math>\sigma</math> must be parameterized. Let <math>\mathbf{q}_{\sigma}(u, v)</math> be a continuous function that maps each point <math>(u,v)</math> from a two-dimensional domain <math>D_{u,v}</math> to a point in <math>\sigma</math>. <math>\mathbf{q}_{\sigma}(u, v)</math> must be continuous and onto. While <math>\mathbf{q}_{\sigma}(u, v)</math> does not necessarily have to be one to one, the parameterization should never "fold back" on itself. The infinitesimal increases in <math>u</math> and <math>v</math> are respectively <math>du</math> and <math>dv</math>. These respectively give rise to the displacements <math>\frac{\partial \mathbf{q}_{\sigma}}{\partial u}du</math> and <math>\frac{\partial \mathbf{q}_{\sigma}}{\partial v}dv</math>. Assuming that the surface's orientation follows the right hand rule with respect to the displacements <math>\frac{\partial \mathbf{q}_{\sigma}}{\partial u}du</math> and <math>\frac{\partial \mathbf{q}_{\sigma}}{\partial v}dv</math>, the surface vector that arises is <math>\mathbf{dS} = (\frac{\partial \mathbf{q}_{\sigma}}{\partial u} \times \frac{\partial \mathbf{q}_{\sigma}}{\partial v})dudv</math>. | ||
| + | |||
| + | In the surface integral <math>\iint_{\mathbf{q} \in \sigma} \mathbf{F}(\mathbf{q}) \cdot \mathbf{dS}</math>, the differential <math>\mathbf{dS}</math> can be replaced with <math>(\frac{\partial \mathbf{q}_{\sigma}}{\partial u} \times \frac{\partial \mathbf{q}_{\sigma}}{\partial v})dudv</math> to get <math>\iint_{(u,v) \in D_{u,v}} \mathbf{F}(\mathbf{q}_{\sigma}(u,v)) \cdot (\frac{\partial \mathbf{q}_{\sigma}}{\partial u} \times \frac{\partial \mathbf{q}_{\sigma}}{\partial v})dudv</math>. | ||
| + | |||
| + | {{DropBox|Example 3| | ||
| + | Consider the problem of computing the surface area of a sphere of radius <math>R</math>. | ||
| + | |||
| + | Center the sphere <math>\sigma</math> on the origin, and using <math>u</math> and <math>v</math> as the parameter variables, the sphere can be parameterized in spherical coordinates via <math>\mathbf{q}_{\sigma}(u,v) = (r = R, \theta = u, \phi = v)</math> where <math>u \in [0, \pi]</math> and <math>v \in [-\pi, +\pi]</math>. The infinitesimal displacements from small changes in the parameters are: | ||
| + | |||
| + | <math>du</math> causes | ||
| + | <math>\frac{\partial\mathbf{q}_{\sigma}}{\partial u}du = (\frac{\partial r}{\partial u}\hat{\mathbf{r}} + r\frac{\partial \theta}{\partial u}\hat{\mathbf{\theta}} + r\sin\theta\frac{\partial \phi}{\partial u}\hat{\mathbf{\phi}})du = (R\hat{\mathbf{\theta}})du</math> | ||
| + | |||
| + | <math>dv</math> causes | ||
| + | <math>\frac{\partial\mathbf{q}_{\sigma}}{\partial v}dv = (\frac{\partial r}{\partial v}\hat{\mathbf{r}} + r\frac{\partial \theta}{\partial v}\hat{\mathbf{\theta}} + r\sin\theta\frac{\partial \phi}{\partial v}\hat{\mathbf{\phi}})dv = (R\sin(u)\hat{\mathbf{\phi}})dv</math> | ||
| + | |||
| + | The infinitesimal surface vector is hence <math>\mathbf{dS} = (R\hat{\mathbf{\theta}})du \times (R\sin(u)\hat{\mathbf{\phi}})dv = (R^2\sin(u)\hat{\mathbf{r}})dudv</math>. While not important to this example, note how the parameterization was chosen so that the surface vector points outwards. The area is <math>|\mathbf{dS}| = R^2\sin(u)dudv</math>. | ||
| + | |||
| + | The total surface area is hence: | ||
| + | |||
| + | <math>\iint_{\mathbf{q} \in \sigma} |\mathbf{dS}| = \int_{u = 0}^{\pi}\int_{v = -\pi}^{+\pi} R^2\sin(u)dvdu = 2\pi R^2\int_{u = 0}^{\pi} \sin(u)du | ||
| + | = 2\pi R^2 (-\cos(u)\bigg|_{u=0}^{\pi}) = 4\pi R^2</math> | ||
| + | |hidden = hidden}} | ||
| + | |||
==Resources== | ==Resources== | ||
* [https://math.libretexts.org/Bookshelves/Calculus/Supplemental_Modules_(Calculus)/Vector_Calculus/4%3A_Integration_in_Vector_Fields/4.3%3A_Line_Integrals Line Integrals], Mathematics LibreTexts | * [https://math.libretexts.org/Bookshelves/Calculus/Supplemental_Modules_(Calculus)/Vector_Calculus/4%3A_Integration_in_Vector_Fields/4.3%3A_Line_Integrals Line Integrals], Mathematics LibreTexts | ||
| + | * [https://en.wikibooks.org/wiki/Calculus/Vector_calculus Vector Calculus], Wikibooks: Calculus | ||
* [https://www.khanacademy.org/math/multivariable-calculus/integrating-multivariable-functions/line-integrals/v/introduction-to-the-line-integral Introduction to The Line Integral], Khan Academy | * [https://www.khanacademy.org/math/multivariable-calculus/integrating-multivariable-functions/line-integrals/v/introduction-to-the-line-integral Introduction to The Line Integral], Khan Academy | ||
Revision as of 20:26, 10 October 2021
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear integral are also used; contour integral is used as well, although that is typically reserved for line integrals in the complex plane.
The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on intervals. Many simple formulae in physics, such as the definition of work as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W=\mathbf{F}\cdot\mathbf{s}} , have natural continuous analogues in terms of line integrals, in this case , which computes the work done on an object moving through an electric or gravitational field F along a path .
Contents
Volume, path (line), and surface integrals
Volume Integrals
Volume integrals have already been discussed in the chapter Multivariable calculus, but a brief review is given here for completeness.
Given a scalar field that denotes a density at each specific point, and an arbitrary volume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega \subseteq \R^3} , the total "mass" Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} inside of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega} can be determined by partitioning Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega} into infinitesimal volumes. At each position Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{q} \in \Omega} , the volume of the infinitesimal volume is denoted by the infinitesimal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dV} . This gives rise to the following integral:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M = \iiint_{\mathbf{q} \in \Omega} \rho(\mathbf{q})dV }
Path Integrals
Given any oriented path Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} (oriented means that there is a preferred direction), the differential Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\mathbf{q} = dx\mathbf{i} + dy\mathbf{j} + dz\mathbf{k}} denotes an infinitesimal displacement along Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} in the preferred direction. This differential can be used in various path integrals. Letting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f: \R^3 \to \R} denote an arbitrary scalar field, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{F}: \R^3 \to \R^3} denote an arbitrary vector field, various path integrals include:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\mathbf{q} \in C} f(\mathbf{q})d\mathbf{q}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\mathbf{q} \in C} f(\mathbf{q})|d\mathbf{q}|} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\mathbf{q} \in C} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\mathbf{q} \in C} \mathbf{F}(\mathbf{q})|d\mathbf{q}|} , and many more.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\mathbf{q} \in C} d\mathbf{q}} denotes the total displacement along Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\mathbf{q} \in C} |d\mathbf{q}|} denotes the total length of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} .
Calculating Path Integrals
To compute a path integral, the continuous oriented curve Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} must be parameterized. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{q}_C(t)} will denote the point along Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} indexed by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} from the range Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [t_0, t_1]} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{q}_C(t_0) = \mathbf{q}_0} must be the starting point of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{q}_C(t_1) = \mathbf{q}_1} must be the ending point of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} . As Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} increases, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{q}_C(t)} must proceed along Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} in the preferred direction. An infinitesimal change in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dt} , results in the infinitesimal displacement Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\mathbf{q} = \frac{d\mathbf{q}_C}{dt}dt} along Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} . In the path integral Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\mathbf{q} \in C} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q}} , the differential Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\mathbf{q}} can be replaced with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d\mathbf{q}_C}{dt}dt} to get Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{t = t_0}^{t_1} \mathbf{F}(\mathbf{q}_C(t)) \cdot \frac{d\mathbf{q}_C}{dt}dt}
If a vector field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{F}} denotes a "force field", which returns the force on an object as a function of position, the work performed on a point mass that traverses the oriented curve Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W = \int_{\mathbf{q} \in C} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q}}
Surface Integrals
Given any oriented surface Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} (oriented means that the there is a preferred direction to pass through the surface), an infinitesimal portion of the surface is defined by an infinitesimal area Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |dS|} , and a unit length outwards oriented normal vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{n}} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{n}} has a length of 1 and is perpendicular to the surface of , while penetrating in the preferred direction. The infinitesimal portion of the surface is denoted by the infinitesimal "surface vector": Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{dS} = |dS|\mathbf{n}} . If a vector field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{F}: \R^3 \to \R^3} denotes a flow density, then the flow through the infinitesimal surface portion in the preferred direction is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{F}(\mathbf{q}) \cdot \mathbf{dS}} .
The infinitesimal "surface vector" Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{dS} = \mathbf{n}|dS|} describes the infinitesimal surface element in a manner similar to how the infinitesimal displacement Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\mathbf{q}} describes an infinitesimal portion of a path. More specifically, similar to how the interior points on a path do not affect the total displacement, the interior points on a surface to not affect the total surface vector.
Consider for instance two paths Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_2} that both start at point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} , and end at point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} . The total displacements, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\mathbf{q} \in C_1} d\mathbf{q}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\mathbf{q} \in C_2} d\mathbf{q}} , are both equivalent and equal to the displacement between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} . Note however that the total lengths Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\mathbf{q} \in C_1} |d\mathbf{q}|} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\mathbf{q} \in C_2} |d\mathbf{q}|} are not necessarily equivalent.
Similarly, given two surfaces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_2} that both share the same counter-clockwise oriented boundary Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} , the total surface vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\mathbf{q} \in S_1} \mathbf{dS}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\mathbf{q} \in S_2} \mathbf{dS}} are both equivalent and are a function of the boundary Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} . This implies that a surface can be freely deformed within its boundaries without changing the total surface vector. Note however that the surface areas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\mathbf{q} \in S_1} |\mathbf{dS}|} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\mathbf{q} \in S_2} |\mathbf{dS}|} are not necessarily equivalent.
The fact that the total surface vectors of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_2} are equivalent is not immediately obvious. To prove this fact, let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{F}} be a constant vector field. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_2} share the same boundary, so the flux/flow of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{F}} through Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_2} is equivalent. The flux through Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_1} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_1 = \int_{\mathbf{q} \in S_1} \mathbf{F} \cdot \mathbf{dS} = \mathbf{F} \cdot \int_{\mathbf{q} \in S_1} \mathbf{dS} } , and similarly for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_2} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_2 = \int_{\mathbf{q} \in S_2} \mathbf{F} \cdot \mathbf{dS} = \mathbf{F} \cdot \int_{\mathbf{q} \in S_2} \mathbf{dS} } . Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{F} \cdot \int_{\mathbf{q} \in S_1} \mathbf{dS} = \mathbf{F} \cdot \int_{\mathbf{q} \in S_2} \mathbf{dS} } for every choice of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{F}} , it follows that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\mathbf{q} \in S_1} \mathbf{dS} = \int_{\mathbf{q} \in S_2} \mathbf{dS} } .
The geometric significance of the total surface vector is that each component measures the area of the projection of the surface onto the plane formed by the other two dimensions. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} be a surface with surface vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{S} = S_x\mathbf{i} + S_y\mathbf{j} + S_z\mathbf{k}} . It is then the case that: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_x} is the area of the projection of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} onto the yz-plane; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_y} is the area of the projection of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} onto the xz-plane; and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_z} is the area of the projection of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} onto the xy-plane.
Given an oriented surface Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma} , another important concept is the oriented boundary. The boundary of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma} is an oriented curve Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \partial\Sigma} but how is the orientation chosen? If the boundary is "counter-clockwise" oriented, then the boundary must follow a counter-clockwise direction when the oriented surface normal vectors point towards the viewer. The counter-clockwise boundary also obeys the "right-hand rule": If you hold your right hand with your thumb in the direction of the surface normals (penetrating the surface in the "preferred" direction), then your fingers will wrap around in the direction of the counter-clockwise oriented boundary.
Calculating Surface Integrals
To calculate a surface integral, the oriented surface Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} must be parameterized. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{q}_{\sigma}(u, v)} be a continuous function that maps each point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (u,v)} from a two-dimensional domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_{u,v}} to a point in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{q}_{\sigma}(u, v)} must be continuous and onto. While Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{q}_{\sigma}(u, v)} does not necessarily have to be one to one, the parameterization should never "fold back" on itself. The infinitesimal increases in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} are respectively Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle du} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dv} . These respectively give rise to the displacements Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial \mathbf{q}_{\sigma}}{\partial u}du} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial \mathbf{q}_{\sigma}}{\partial v}dv} . Assuming that the surface's orientation follows the right hand rule with respect to the displacements Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial \mathbf{q}_{\sigma}}{\partial u}du} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial \mathbf{q}_{\sigma}}{\partial v}dv} , the surface vector that arises is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{dS} = (\frac{\partial \mathbf{q}_{\sigma}}{\partial u} \times \frac{\partial \mathbf{q}_{\sigma}}{\partial v})dudv} .
In the surface integral Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iint_{\mathbf{q} \in \sigma} \mathbf{F}(\mathbf{q}) \cdot \mathbf{dS}} , the differential Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{dS}} can be replaced with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\frac{\partial \mathbf{q}_{\sigma}}{\partial u} \times \frac{\partial \mathbf{q}_{\sigma}}{\partial v})dudv} to get Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iint_{(u,v) \in D_{u,v}} \mathbf{F}(\mathbf{q}_{\sigma}(u,v)) \cdot (\frac{\partial \mathbf{q}_{\sigma}}{\partial u} \times \frac{\partial \mathbf{q}_{\sigma}}{\partial v})dudv} .
Resources
- Line Integrals, Mathematics LibreTexts
- Vector Calculus, Wikibooks: Calculus
- Introduction to The Line Integral, Khan Academy
