Department of Mathematics at UTSA - User contributions [en]

MAT1224

2023-06-30T20:38:25Z

Benjamin.sencindiver: /* Topics List */ Ben S - 6-30-23 - minor formatting change to table of contents.

The textbook for this course is
[https://openstax.org/details/books/calculus-volume-2 Calculus (Volume 2) by Gilbert Strang, Edwin Herman, et al.]

A comprehensive list of all undergraduate math courses at UTSA can be found [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ here].

The Wikipedia summary of [https://en.wikipedia.org/wiki/Calculus calculus and its history].

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes

|-

|Week 1

||

<div style="text-align: center;">1.3</div>

||

[[The Fundamental Theorem of Calculus]]

||

* [[Differentiation Rules]] 
* [[Chain Rule|The Chain Rule]] 
* [[Antiderivatives]] 
* [[The Definite Integral]] 

||
*
* Use the Fundamental Theorem of Calculus, Part 1, to evaluate derivatives of integrals.
* Use the Fundamental Theorem of Calculus, Part 2, to evaluate definite integrals.
* Explain the relationship between differentiation and integration.

|-

|Week 1

||

<div style="text-align: center;">1.5</div>

||

[[Integration by Substitution]]

||

* [[Differentiation Rules]] 
* [[Linear Approximations and Differentials|Differentials]] 
* [[Antiderivatives]] 
* [[The Definite Integral]] 
* [[The Fundamental Theorem of Calculus]] 

||

* Recognize when to use integration by substitution.
* Use substitution to evaluate indefinite integrals.
* Use substitution to evaluate definite integrals.

|-

|Week 2

||

<div style="text-align: center;">2.1</div>

||

[[Area between Curves]]

||

* [[Toolkit Functions|Graphing Elementary Functions]] 
* [[The Definite Integral]] 
* [[The Fundamental Theorem of Calculus]] 
* [[Integration by Substitution]] 

||

* Determine the area of a region between two curves by integrating with respect to the independent variable.
* Find the area of a compound region.
* Determine the area of a region between two curves by integrating with respect to the dependent variable.

|-

|Week 2

||

<div style="text-align: center;">2.2</div>

||

[[Determining Volumes by Slicing]]

||

* [[Areas of basic shapes]] 
* [[Volume of a cylinder]] 
* [[Toolkit Functions|Graphing elementary functions]] 
* [[The Fundamental Theorem of Calculus]] 
* [[Integration by Substitution]] 

||

* Determine the volume of a solid by integrating a cross-section (the slicing method).
* Find the volume of a solid of revolution using the disk method.
* Find the volume of a solid of revolution with a cavity using the washer method.

|-

|Week 3

||

<div style="text-align: center;">2.3</div>

||

[[Volumes of Revolution, Cylindrical Shells]]

||

* [[Toolkit Functions|Graphing elementary functions]] 
* [[Determining Volumes by Slicing]] 
* [[The Fundamental Theorem of Calculus]] 
* [[Integration by Substitution]] 

||

* Calculate the volume of a solid of revolution by using the method of cylindrical shells.
* Compare the different methods for calculating a volume of revolution.

|-

|Week 3
||

<div style="text-align: center;">2.4</div>

||

[[Arc Length and Surface Area]]

||

* [[Differentiation Rules]] 
* [[The Fundamental Theorem of Calculus]] 
* [[Integration by Substitution]] 

||

* Determine the length of a plane curve between two points.
* Find the surface area of a solid of revolution.

|-

|Week 4

||

<div style="text-align: center;">2.5</div>

||

[[Physical Applications]]

||

* [[Areas of basic shapes]] 
* [[Volume of a cylinder]] 
* [[Basic Physics (Mass, Force, Work, Newton's Second Law, Hooke's Law)]] 
* [[The Fundamental Theorem of Calculus]] 
* [[Integration by Substitution]] 

||

* Determine the mass of a one-dimensional object from its linear density function.
* Determine the mass of a two-dimensional circular object from its radial density function.
* Calculate the work done by a variable force acting along a line.
* Calculate the work done in stretching/compressing a spring.
* Calculate the work done in lifting a rope/cable.
* Calculate the work done in pumping a liquid from one height to another.
* Find the hydrostatic force against a submerged vertical plate.

|-

|Week 4

||

<div style="text-align: center;">2.6</div>

||

[[Moments and Center of Mass]]

||

* [[Toolkit Functions|Graphing elementary functions]] 
* [[The Fundamental Theorem of Calculus]] 
* [[Integration by Substitution]] 

||

* Find the center of mass of objects distributed along a line.
* Find the center of mass of objects distributed in a plane.
* Locate the center of mass of a thin plate.
* Use symmetry to help locate the centroid of a thin plate.

|-

|Week 5

||

<div style="text-align: center;">3.1</div>

||

[[Integration by Parts]]

||

* [[Differentiation Rules]] 
* [[Linear Approximations and Differentials|Differentials]] 
* [[The Fundamental Theorem of Calculus]] 
* [[Integration by Substitution]] 

||

* Recognize when to use integration by parts.
* Use the integration-by-parts formula to evaluate indefinite integrals.
* Use the integration-by-parts formula to evaluate definite integrals.
* Use the tabular method to perform integration by parts.
* Solve problems involving applications of integration using integration by parts.

|-

|Week 5

||

<div style="text-align: center;">3.2</div>

||

[[Trigonometric Integrals]]

||

* [[Trigonometric Functions]] 
* [[Properties of the Trigonometric Functions|Trigonometric Identities]] 
* [[The Fundamental Theorem of Calculus]] 
* [[Integration by Substitution]] 
* [[Integration by Parts]] 

||

* Evaluate integrals involving products and powers of sin(x) and cos(x).
* Evaluate integrals involving products and powers of sec(x) and tan(x).
* Evaluate integrals involving products of sin(ax), sin(bx), cos(ax), and cos(bx).
* Solve problems involving applications of integration using trigonometric integrals.

|-

|Week 6

||

<div style="text-align: center;">3.3</div>

||

[[Trigonometric Substitution]]

||

* [[Completing the Square]] 
* [[Trigonometric Functions]] 
* [[Properties of the Trigonometric Functions|Trigonometric Identities]] 
* [[Integration by Substitution]] 
* [[Integration by Parts]] 
* [[Trigonometric Integrals]] 

||

* Evaluate integrals involving the square root of a sum or difference of two squares.
* Solve problems involving applications of integration using trigonometric substitution.

|-

|Week 6

||

<div style="text-align: center;">3.4</div>

||

[[Partial Fractions]]

||

* [[Factoring Polynomials]] 
* [[Completing the Square]] 
* [[Dividing Polynomials|Long Division of Polynomials]] 
* [[Systems of Linear Equations]] 
* [[Antiderivatives]] 
* [[Integration by Substitution]] 

||

* Integrate a rational function whose denominator is a product of linear and quadratic polynomials.
* Recognize distinct linear factors in a rational function.
* Recognize repeated linear factors in a rational function.
* Recognize distinct irreducible quadratic factors in a rational function.
* Recognize repeated irreducible quadratic factors in a rational function.
* Solve problems involving applications of integration using partial fractions.

|-

|Week 7

||

<div style="text-align: center;">3.7</div>

||

[[Improper Integrals]]

||

* [[The Fundamental Theorem of Calculus]] 
* [[Integration by Substitution]] 
* [[Integration by Parts]] 
* [[Trigonometric Integrals]] 
* [[Trigonometric Substitution]] 
* [[Partial Fractions]] 
* [[The Limit Laws]] 
* [[Limits at Infinity and Asymptotes| Limits at Infinity]] 
* [[L’Hôpital’s Rule]] 

||

* Recognize improper integrals and determine their convergence or divergence.
* Evaluate an integral over an infinite interval.
* Evaluate an integral over a closed interval with an infinite discontinuity within the interval.
* Use the comparison theorem to determine whether an improper integral is convergent or divergent.

|-

|Week 8

||

<div style="text-align: center;">4.3</div>

||

[[Separation of Variables]]

||

* [[Factoring Polynomials]] 
* [[Exponential Properties]] 
* [[Logarithmic Properties]] 
* [[Linear Approximations and Differentials|Differentials]] 
* [[Initial Value Problem]] 
* [[Integration by Substitution]] 
* [[Integration by Parts]] 
* [[Trigonometric Integrals]] 
* [[Trigonometric Substitution]] 
* [[Partial Fractions]] 

||

* Recognize separable differential equations.
* Use separation of variables to solve a differential equation.
* Develop and analyze elementary mathematical models.

|-

|Week 8

||

<div style="text-align: center;">4.5</div>

||

[[First-Order Linear Equations]]

||

* [[Separation of Variables]] 

||

* Write a first-order linear differential equation in standard form.
* Find an integrating factor and use it to solve a first-order linear differential equation.
* Solve applied problems involving first-order linear differential equations.

|-

|Week 9

||

<div style="text-align: center;">5.1</div>

||

[[Sequences]]

||

* [[The Limit Laws| The Limit Laws and Squeeze Theorem]] 
* [[Limits at Infinity and Asymptotes| Limits at Infinity]] 
* [[L’Hôpital’s Rule]] 
* [[Derivatives and the Shape of a Graph| Increasing and Decreasing Functions]] 

||

* Find a formula for the general term of a sequence.
* Find a recursive definition of a sequence.
* Determine the convergence or divergence of a given sequence.
* Find the limit of a convergent sequence.
* Determine whether a sequence is bounded and/or monotone.
* Apply the Monotone Convergence Theorem.

|-

|Week 10

||

<div style="text-align: center;">5.2</div>

||

[[Infinite Series]]

||

* [[Sigma notation]] 
* [[Sequences]] 
* [[Partial Fractions]] 

||

* Write an infinite series using sigma notation.
* Find the nth partial sum of an infinite series.
* Define the convergence or divergence of an infinite series.
* Identify a geometric series.
* Apply the Geometric Series Test.
* Find the sum of a convergent geometric series.
* Identify a telescoping series.
* Find the sum of a telescoping series.

|-

|Week 10

||

<div style="text-align: center;">5.3</div>

||

[[The Divergence and Integral Tests]]

||

* [[The Limit Laws]] 
* [[Limits at Infinity and Asymptotes| Limits at Infinity]] 
* [[Continuity]] 
* [[Derivatives and the Shape of a Graph| Increasing and Decreasing Functions]] 
* [[L’Hôpital’s Rule]] 
* [[Improper Integrals]] 

||

* Use the Divergence Test to determine whether a series diverges.
* Use the Integral Test to determine whether a series converges or diverges.
* Use the p-Series Test to determine whether a series converges or diverges.
* Estimate the sum of a series by finding bounds on its remainder term.

|-

|Week 11

||

<div style="text-align: center;">5.4</div>

||

[[Comparison Tests]]

||

* [[Limits at Infinity and Asymptotes|Limits at Infinity]] 
* [[Derivatives and the Shape of a Graph|Increasing and Decreasing Functions]] 
* [[L’Hôpital’s Rule]] 
* [[Infinite Series|The Geometric Series Test]] 
* [[The Divergence and Integral Tests|The p-Series Test]] 

||

* Use the Direct Comparison Test to determine whether a series converges or diverges.
* Use the Limit Comparison Test to determine whether a series converges or diverges.

|-

|Week 11

||

<div style="text-align: center;">5.5</div>

||

[[Alternating Series]]

||

* [[Limits at Infinity and Asymptotes|Limits at Infinity]] 
* [[Derivatives and the Shape of a Graph|Increasing and Decreasing Functions]] 
* [[L’Hôpital’s Rule]] 
* [[Infinite Series|The Geometric Series Test]] 
* [[The Divergence and Integral Tests|The p-Series Test]] 
* [[Comparison Tests]] 

||

* Use the Alternating Series Test to determine the convergence of an alternating series.
* Estimate the sum of an alternating series.
* Explain the meaning of absolute convergence and conditional convergence.

|-

|Week 12

||

<div style="text-align: center;">5.6</div>

||

[[Ratio and Root Tests]]

||

* [[Factorials]] 
* [[Limits at Infinity and Asymptotes|Limits at Infinity]] 
* [[L’Hôpital’s Rule]] 

||

* Use the Ratio Test to determine absolute convergence or divergence of a series.
* Use the Root Test to determine absolute convergence or divergence of a series.
* Describe a strategy for testing the convergence or divergence of a series.

|-

|Week 12

||

<div style="text-align: center;">6.1</div>

||

[[Power Series and Functions]]

||

* [[Infinite Series|The Geometric Series Test]] 
* [[The Divergence and Integral Tests]] 
* [[Comparison Tests]] 
* [[Alternating Series]] 
* [[Ratio and Root Tests]] 

||

* Identify a power series.
* Determine the interval of convergence and radius of convergence of a power series.
* Use a power series to represent certain functions.

|-

|Week 13

||

<div style="text-align: center;">6.2</div>

||

[[Properties of Power Series]]

||

* [[Differentiation Rules]] 
* [[Antiderivatives]] 
* [[The Fundamental Theorem of Calculus]] 
* [[Power Series and Functions]] 

||

* Combine power series by addition or subtraction.
* Multiply two power series together.
* Differentiate and integrate power series term-by-term.
* Use differentiation and integration of power series to represent certain functions as power series.

|-

|Week 14

||

<div style="text-align: center;">6.3</div>

||

[[Taylor and Maclaurin Series]]

||

* [[The Derivative as a Function|Higher-Order Derivatives]] 
* [[Power Series and Functions]] 
* [[Properties of Power Series]] 

||

* Find a Taylor or Maclaurin series representation of a function.
* Find the radius of convergence of a Taylor Series or Maclaurin series.
* Finding a Taylor polynomial of a given order for a function.
* Use Taylor's Theorem to estimate the remainder for a Taylor series approximation of a given function.

|-

|Week 15

||

<div style="text-align: center;">7.1</div>

||

[[Parametric Equations]]

||

* [[Toolkit Functions|Sketching Elementary Functions]] 
* [[Equation of a Circle]] 
* [[Equation of an Ellipse]] 
* [[Trigonometric Functions]] 
* [[Properties of the Trigonometric Functions|Trigonometric Identities]] 

||

* Plot a curve described by parametric equations.
* Convert the parametric equations of a curve into the form y=f(x) by eliminating the parameter.
* Recognize the parametric equations of basic curves, such as a line and a circle.

|-

|Week 15

||

<div style="text-align: center;">7.2</div>

||

[[The Calculus of Parametric Equations]]

||

* [[Parametric Equations]] 
* [[Differentiation Rules]] 
* [[The Derivative as a Function|Higher-Order Derivatives]] 
* [[The Fundamental Theorem of Calculus]] 
* [[Integration by Substitution]] 
* [[Integration by Parts]] 

||

* Find the slope of the tangent line to a parametric curve at a point.
* Use the second derivative to determine the concavity of a parametric curve at a point.
* Determine the area bounded by a parametric curve.
* Determine the arc length of a parametric curve.
* Determine the area of a surface obtained by rotating a parametric curve about an axis.

|}

Talk:MAT1224

2023-06-30T20:34:30Z

Benjamin.sencindiver: Ben S - 06-30-23 - Minor formatting change on table of contents

===Remarks to Course Correction===
* Systems of Linear Equations is a prereq within MAT 1224 and MAT 3613, but it is not mentioned within MAT 1073. Though, within 1073 they mention as a topic Systems of Equations of Three Variables. Maybe add a general lesson of n variable systems or a specific lesson on systems of linear equations.
* Ben S - 06-30-23 - Minor Adjustment to formatting in table of contents page to make all prerequisites unbolded.

Polar Coordinates

2023-06-21T18:53:40Z

Benjamin.sencindiver: Ben - 6/21/23 - fixing typos (replacing 3 cases of \varphi with \phi).

[[Image:Polar graph paper.svg|thumb|right|300px|A polar grid with several angles labeled in degrees]]
The '''polar coordinate system''' is a two-dimensional coordinate system in which each point on a plane is determined by an angle and a distance. The polar coordinate system is especially useful in situations where the relationship between two points is most easily expressed in terms of angles and distance; in the more familiar Cartesian coordinate system or rectangular coordinate system, such a relationship can only be found through trigonometric formulae.

As the coordinate system is two-dimensional, each point is determined by two polar coordinates: the radial coordinate and the angular coordinate. The radial coordinate (usually denoted as <math>r</math>) denotes the point's distance from a central point known as the ''pole'' (equivalent to the ''origin'' in the Cartesian system). The angular coordinate (also known as the polar angle or the azimuth angle, and usually denoted by <math>\theta</math> or <math>t</math>) denotes the positive or anticlockwise (counterclockwise) angle required to reach the point from the 0° ray or ''polar axis'' (which is equivalent to the positive <math>x</math>-axis in the Cartesian coordinate plane).

==Plotting points with polar coordinates==
[[Image:CircularCoordinates.png|thumb|250px|The points (3,60°) and (4,210°) on a polar coordinate system]]
Each point in the polar coordinate system can be described with the two polar coordinates, which are usually called <math>r</math> (the radial coordinate) and θ (the angular coordinate, polar angle, or azimuth angle, sometimes represented as <math>\phi</math> or <math>t</math>). The <math>r</math> coordinate represents the radial distance from the pole, and the θ coordinate represents the anticlockwise (counterclockwise) angle from the <math>0^\circ</math> ray (sometimes called the polar axis), known as the positive <math>x</math>-axis on the Cartesian coordinate plane.

For example, the polar coordinates <math>(3,60^\circ)</math> would be plotted as a point 3 units from the pole on the <math>60^\circ</math> ray. The coordinates <math>(-3,240^\circ)</math> would also be plotted at this point because a negative radial distance is measured as a positive distance on the opposite ray (the ray reflected about the origin, which differs from the original ray by <math>180^\circ</math>).

One important aspect of the polar coordinate system, not present in the Cartesian coordinate system, is that a single point can be expressed with an infinite number of different coordinates. This is because any number of multiple revolutions can be made around the central pole without affecting the actual location of the point plotted. In general, the point <math>(r,\theta)</math> can be represented as <math>(r,\theta\pm360^\circ k)</math> or <math>(-r,\theta\pm(2k+1)180^\circ)</math> , where <math>k</math> is any integer.

The arbitrary coordinates <math>(0,\theta)</math> are conventionally used to represent the pole, as regardless of the θ coordinate, a point with radius 0 will always be on the pole. To get a unique representation of a point, it is usual to limit <math>r</math> to negative and non-negative numbers <math>r\ge0</math> and <math>\theta</math> to the interval <math>[0,360^\circ)</math> or <math>(-180^\circ,180^\circ]</math> (or, in radian measure, <math>[0,2\pi)</math> or <math>(-\pi,\pi]</math>).

Angles in polar notation are generally expressed in either degrees or radians, using the conversion <math>2\pi\ \text{rad}=360^\circ</math> . The choice depends largely on the context. Navigation applications use degree measure, while some physics applications (specifically rotational mechanics) and almost all mathematical literature on calculus use radian measure.

===Converting between polar and Cartesian coordinates===
[[Image:Polar to cartesian.svg|right|thumb|250px|A diagram illustrating the conversion formulae]]
The two polar coordinates <math>r,\theta</math> can be converted to the Cartesian coordinates <math>x,y</math> by using the trigonometric functions sine and cosine:
:<math>x=r\cos(\theta)</math>
:<math>y=r\sin(\theta)</math>

while the two Cartesian coordinates <math>x,y</math> can be converted to polar coordinate <math>r</math> by
:<math>r=\sqrt{x^2+y^2}</math> (by a simple application of the Pythagorean theorem).

To determine the angular coordinate <math>\theta</math> , the following two ideas must be considered:
*For <math>r=0</math> , <math>\theta</math> can be set to any real value.
*For <math>r\ne0</math> , to get a unique representation for <math>\theta</math> , it must be limited to an interval of size <math>2\pi</math> . Conventional choices for such an interval are <math>[0,2\pi)</math> and <math>(-\pi,\pi]</math> .

To obtain <math>\theta</math> in the interval <math>[0,2\pi)</math> , the following may be used (<math>\arctan</math> denotes the inverse of the tangent function):
:<math>\theta=
\begin{cases}
\arctan\left(\frac{y}{x}\right)&\text{if }x>0\text{ and }y\ge0\\
\arctan\left(\frac{y}{x}\right)+2\pi&\text{if }x>0\text{ and }y<0\\
\arctan\left(\frac{y}{x}\right)+\pi&\text{if }x<0\\
\frac{\pi}{2}&\text{if }x=0\text{ and }y>0\\
\frac{3\pi}{2}&\text{if }x=0\text{ and }y<0
\end{cases}</math>

To obtain <math>\theta</math> in the interval <math>(-\pi,\pi]</math> , the following may be used:

:<math>\theta=
\begin{cases}
\arctan\left(\frac{y}{x}\right)&\text{if }x>0\\
\arctan\left(\frac{y}{x}\right)+\pi&\text{if }x<0\text{ and }y\ge0\\
\arctan\left(\frac{y}{x}\right)-\pi&\text{if }x<0\text{ and }y<0\\
\frac{\pi}{2}&\text{if }x=0\text{ and }y>0\\
-\frac{\pi}{2}&\text{if }x=0\text{ and }y<0
\end{cases}</math>

One may avoid having to keep track of the numerator and denominator signs by use of the atan2 function, which has separate arguments for the numerator and the denominator.

==Polar equations==
The equation defining an algebraic curve expressed in polar coordinates is known as a ''polar equation''. In many cases, such an equation can simply be specified by defining <math>r</math> as a function of <math>\theta</math> . The resulting curve then consists of points of the form <math>\big(r(\theta),\theta\big)</math> and can be regarded as the graph of the polar function <math>r</math> .

Different forms of symmetry can be deduced from the equation of a polar function <math>r</math> . If <math>r(-\theta)=r(\theta)</math> the curve will be symmetrical about the horizontal <math>(0^\circ/180^\circ)</math> ray, if <math>r(\pi-\theta)=r(\theta)</math> it will be symmetric about the vertical <math>(90^\circ/270^\circ)</math> ray, and if <math>r(\theta-\alpha^\circ)=r(\theta)</math> it will be rotationally symmetric <math>\alpha^\circ</math> counterclockwise about the pole.

Because of the circular nature of the polar coordinate system, many curves can be described by a rather simple polar equation, whereas their Cartesian form is much more intricate. Among the best known of these curves are the polar rose, Archimedean spiral, lemniscate, limaçon, and cardioid.

For the circle, line, and polar rose below, it is understood that there are no restrictions on the domain and range of the curve.

===Circle===
[[Image:circle_r=1.PNG|thumb|right|A circle with equation <math>r(\theta)=1</math>]]
The general equation for a circle with a center at <math>(r_0,\phi)</math> and radius <math>a</math> is
:<math>r^2-2rr_0\cos(\theta-\phi)+r_0^2=a^2</math>

This can be simplified in various ways, to conform to more specific cases, such as the equation
:<math>r(\theta)=a</math>

for a circle with a center at the pole and radius <math>a</math> .

===Line===
''Radial'' lines (those running through the pole) are represented by the equation
:<math>\theta=\phi</math>
where <math>\phi</math> is the angle of elevation of the line; that is, <math>\phi=\arctan(m)</math> where <math>m</math> is the slope of the line in the Cartesian coordinate system. The non-radial line that crosses the radial line <math>\theta=\phi</math> perpendicularly at the point <math>(r_0,\phi)</math> has the equation
:<math>r(\theta)=r_0\sec(\theta-\phi)</math>

===Polar rose===
[[Image:rose_r=2sin(4theta).PNG|thumb|right|A polar rose with equation <math>r(\theta)=2\sin(4\theta)</math>]]
A polar rose is a famous mathematical curve that looks like a petaled flower, and that can be expressed as a simple polar equation,
:<math>r(\theta)=a\cos(k\theta+\phi_0)</math>

for any constant <math>\phi_0</math> (including 0). If <math>k</math> is an integer, these equations will produce a <math>k</math>-petaled rose if <math>k</math> is odd, or a <math>2k</math>-petaled rose if <math>k</math> is even. If <math>k</math> is rational but not an integer, a rose-like shape may form but with overlapping petals. Note that these equations never define a rose with 2, 6, 10, 14, etc. petals. The variable <math>a</math> represents the length of the petals of the rose.

===Archimedean spiral===
[[Image:Archimedian_spiral.PNG|thumb|right|One arm of an Archimedean spiral with equation <math>r(\theta)=\theta</math> for <math>\theta\in(0,6\pi)</math>]]
The Archimedean spiral is a famous spiral that was discovered by Archimedes, which also can be expressed as a simple polar equation. It is represented by the equation
:<math>r(\theta)=a+b\theta</math>

Changing the parameter <math>a</math> will turn the spiral, while <math>b</math> controls the distance between the arms, which for a given spiral is always constant. The Archimedean spiral has two arms, one for <math>\theta>0</math> and one for <math>\theta<0</math> . The two arms are smoothly connected at the pole. Taking the mirror image of one arm across the <math>90^\circ/270^\circ</math> line will yield the other arm. This curve is notable as one of the first curves, after the conic sections, to be described in a mathematical treatise, and as being a prime example of a curve that is best defined by a polar equation.

===Conic sections===
[[Image:Elps-slr.svg|thumb|right|250px|Ellipse, showing semi-latus rectum]]
A conic section with one focus on the pole and the other somewhere on the <math>0^\circ</math> ray (so that the conic's semi-major axis lies along the polar axis) is given by:
:<math>r=\frac{\ell}{1+\epsilon\cos(\theta)}</math>
where <math>\epsilon</math> is the eccentricity and <math>\ell</math> is the semi-latus rectum (the perpendicular distance at a focus from the major axis to the curve).
#If <math>\epsilon>1</math> , this equation defines a hyperbola.
#If <math>\epsilon=1</math> , it defines a parabola.
#If <math>0<\epsilon<1</math> , it defines an ellipse. The special case <math>\epsilon=0</math> of the latter results in a circle of radius <math>\ell</math> .

==Resources==
* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Polar%20Coordinates/Esparza%201093%20Notes%205.1.pdf Polar Coordinates]. Written notes created by Professor Esparza, UTSA.
* [https://www.youtube.com/watch?v=aSdaT62ndYE Polar Coordinates Basic Introduction, Conversion to Rectangular, How to Plot Points, Negative R Value ] Video by The Organic Chemistry Tutor 2017
* [https://www.youtube.com/watch?v=r0fv9V9GHdo&t=69s Polar Coordinates - The Basics ] Video by patrickJMT 2018

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikibooks.org/wiki/Calculus/Polar_Introduction Polar Introduction, Wikibooks] under a CC BY-SA license

MAT2214

2023-06-21T16:10:38Z

Benjamin.sencindiver: /* Topics List */ Improving Consistency of Formatting

The textbook for this course is
[https://openstax.org/details/books/calculus-volume-3 Calculus (Volume 3) by Gilbert Strang, Edwin Herman, et al.]

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-

|Week 1

||
1.1
||

[[Polar Coordinates]]

||
* [[Trigonometric Functions: Unit Circle Approach]]
* [[Inverse Trigonometric Functions]]
||
* Plot points using polar coordinates and find several polar coordinates of a single point
* Convert polar coordinates to rectangular coordinates and vice versa
* Transform equations from polar form to rectangular form and vice versa
|-

|Week 1

||
1.2
||

[[Three-Dimensional Coordinate Systems]]

||

* [[Two-dimensional coordinate systems]]
* [[Solving Equations and Inequalities| Algebraic Expressions]]

||

* Three-dimensional coordinate systems.
* Distance Formula in Space.
* Standard Equation for a Sphere.
|-

|Weeks 1/2

||
2.1
||

[[Vectors in The Plane, Space]]

||

* [[Linear Equations|Line Segments]]
* [[Distance Formula| Distance Formula]]

||

* Vector Algebra Operations
* The Magnitude of a vector
* Unit Vectors
* The Midpoint of a Line Segment
* The Vector projection
|-

|Week 2

||
2.3
||

[[The Dot Product]]

||

* [[Trigonometric Functions|Basic Trig Functions]]
* [[Vectors]]

||
* Definition of Dot Product
* Properties of Dot Product
* Angle between vectors
* Orthogonal vectors

|-

|Week 2

||
2.4
||

[[The Cross Product]]

||

* [[Trigonometric Functions|Basic Trig Functions]]
* [[Determinants]]
* [[Vectors]]

||

* Definition of Cross Product
* Properties of the cross product
* Area of a parallelogram
* Cross product as a determinant

|-

|Week 3

||
2.5
||

[[Equations of Lines, Planes and Surfaces in Space]]

||

* [[The Dot Product]]
* [[The Cross Product]]
* [[Quadratic Functions]]
* [[Parametric Equations]]

||

* Write the vector, parametric equation of a line through a given point in a given direction, and a line through two given points.
* Find the distance from a point to a given line.
* Write the equation of a plane through a given point with a given normal, and a plane through three given points.
* Find the distance from a point to a given plane.

|-

|Week 3

||
2.6
||

[[Equations of Lines, Planes and Surfaces in Space|Cylinders and Quadratic Surfaces]]

||

* [[Quadratic Functions]]
* [[Parametric Equations]]
* [[Conics]]

||

* Find equations for cylinders that are generated by rotating lines that are parallel to a plane
* Understand basic quadratic surfaces
* Understand general quadratic surfaces

|-

|Weeks 3/4

||
3.1, 3.2
||

[[Curves in Space and Vector-Valued Functions]]

||

* [[Parametric Equations]]
* [[Vectors]]
* [[The Derivative as a Function]]
* [[The Limit of a Function]]
* [[Continuity]]
* [[The Dot Product]]
* [[The Cross Product]]

||

* Vector functions
* Limits of vector functions
* Continuity of vector functions
* Differentiation rules for vector functions
* Curves and paths in space

|-

|Week 4

||
3.3
||

[[Arc Length]]

||

* [[Distance Formula| The Length of a Line Segment]]
* [[Curves in Space and Vector-Valued Functions|Vector Functions]]
* [[Line Integrals|Integrals of Vector Functions]], [[Derivatives of Vector Functions]]

||

* The arc Length of a vector function
* Arc length parameterization

|-

|Weeks 4/5

||
3.4
||

[[Motion in Space]]

||
* [[Vectors]]
* [[Parametric Equations]]
* [[The Cross Product]]
* [[Derivatives of Vector Functions]]
||
* The Unit tangent vector
* The curvature
* The Principal Unit Normal Vector
* The Binormal Vector
* The tangential and normal components of acceleration
* The Torsion

|-

|Week 5/6

||
4.1
||

[[Functions of Several Variables]]

||

* [[Domain of a Function]]
* [[Range of a Function]]
* [[Solving Equations and Inequalities]]
* [[Graphs| Graphing a Function]]

||
* Functions of two variables
* Functions of three variables
* Domain and range of multivariable functions
* Bounded regions
* Graphs and level curves of two variable functions
* Level surfaces of three variable functions
|-

|Week 6

||
4.2
||

[[Limit and Continuity of Function of Several Variables]]

||

* [[Continuity]]
* [[The Limit Laws]]
* [[Composition of Functions]]
* [[The Dot Product]]

||

* Limits of functions of two variables
* Limits of functions of more than two variables
* Properties of limits of functions of several variables
* Two path test of non-existing of a limit
* Continuity for functions of several variables
* Continuity of composition
* Extreme values on closed and bounded domains
|-

|Week 6

||
4.3
||

[[Partial Derivatives]]

||

* [[The Derivative as a Function|The first]] and [[The Second Derivative|second derivative]] of a function
* [[Limit and Continuity of Function of Several Variables]]

||
* Partial derivatives for functions of two variables
* Partial derivatives for functions of more than two variables
* Partial derivatives and continuity
* Second order partial derivatives
* Mixed derivative theorem
|-

|Week 7

||
4.4
||

[[Directional Derivatives and Gradient Vectors]]

||
* [[Trigonometric Functions]]
* [[Vectors, Unit Vectors]]
* [[Partial Derivatives]]
* [[Gradients]]
* [[The Dot Product]]
||
* Directional derivatives for functions of two variables
* Gradients
* Properties of directional derivatives
* Tangents to level curves
* Directional derivatives for functions of three variables
|-

|Week 7

||
4.5
||
[[Tangent Plane]],
[[Differentiability]]

||
* [[Partial Derivatives]]
* [[Parametric Equations]] of Lines
* [[Equations of Lines, Planes and Surfaces in Space]]

||
* Determine the equation of a plane tangent to a given surface at a point
* Determine the parametric equation of a normal line to a given surface at a point
* The linear approximation of a function of two variables at a point
* The definition of differentiability for a function of two variables
* Differentiability implies Continuity
* Continuity of First Partial Derivatives implies Differentiability
* The definition of total differentiability for a function of two variables
* Use the total differential to approximate the change in a function of two variables
|-

|Week 7

||
4.6
||

[[The Chain Rule for Functions of more than One Variable]]

||
* [[Differentiation Rules]]
* [[The Chain Rule]]
* [[Partial Derivatives]]

||
* Chain rule for functions of one independent variable and several intermediate variables.
* Chain rule for functions of two independent variable and several intermediate variables.
* Method for implicit differentiation.
* The general chain rule for functions of several independent variables
|-
|Week 8

||
4.7
||

[[Maxima and Minima Problems]]

||
* [[Extreme values on closed and bounded domains]]
* [[Partial Derivatives]]
* [[Maxima and Minima|Maxima, Minima and Critical Points of a Function]]
* [[Limit and Continuity of Function of Several Variables]]

||
* The derivative test for local extreme values
* Extreme values on closed and bounded domains
* Critical points and saddle points for functions of two variables
* Second derivative test for local extreme values
* Absolute maxima and minima on closed and bounded regions
|-
|Week 8/9

||
4.8
||

[[Lagrange Multipliers]]

||

* [[Partial Derivatives]]
* [[Critical Points of a Function]]

||
* Lagrange Multipliers with One Constraint
* Lagrange Multipliers with Two Constraints
|-

|Week 9/10

||
5.1
||

[[Multiple Integrals|Double Integrals over Rectangular Regions]]

||

* [[Approximating Areas]]
* [[The Definite Integral|Limits of Riemann Sums]]

||
* Double Integral is the limit of Double Sums.
* Double Integrals over Rectangular Regions.
* Interated Integrals.
* Fubini's Theorem (part 1).
|-

|Week 10

||
5.2
||

[[Multiple Integrals|Double Integrals over General Regions]]

||

* [[Continuity]]
* [[Determining Volumes by Slicing]]
* [[Multiple Integrals|Double and Iterated Integrals over Rectangular regions]]

||
* Double integrals over bounded, general regions.
* Properties of double Integrals.
* Fubini's theorem (part 2)
* Changing the order of Integration.
* Calculating Volumes, Areas and Average Values
|-

|Week 11

||
5.3
||

[[Multiple Integrals|Double Integrals in Polar Coordinates]]

||

* [[Multiple Integrals|Double Integrals over General Regions]]
* [[Polar Coordinates]]

||
* Double Integrals over rectangular polar regions.
* Double Integrals over general polar regions.
* Changing Cartesian Integrals into Polar Integrals.
* Using Double Integrals in Polar Coordinates to find Volumes, Areas.
|-

|Week 11

||
5.4
||

[[Multiple Integrals|Triple Integrals in Rectangular Coordinates]]

||

* [[Multiple Integrals|Double Integrals]]
* [[Multiple Integrals|Area by Double Integration]]
* [[Change of Variables]]

||

* Triple Integrals over general bounded regions.
* Finding Volumes by evaluating Triple Integrals.
* Average value of a function in space.
* Changing Integration Order and Coordinate systems.
|-

|Week 12

||
5.5
||

[[Multiple Integrals|Triple Integrals in Cylindrical and Spherical Coordinates]]

||

* [[Multiple Integrals|Double Integrals in Polar Form]]
* [[Multiple Integrals|Triple Integrals in Rectangular Coordinates]]

||

* Integrations in Cylindrical Coordinates.
* Equations relating rectangular and cylindrical coordinates.
* Changing Cartesian integrations into Cylindrical integrations.
* Integrations in Spherical coordinates.
* Equations relating spherical coordinates to Cartesian and cylindrical coordinates.
* Changing Cartesian integrations into Cylindrical integrations.
|-

|Week 13

||
5.6
||

[[Multiple Integrals|Applications of Multiple Integrals]]

||

* [[Multiple Integrals|Double Integral]]
* [[Multiple Integrals|Triple Integrals]]

||

* Finding Masses, Moments, Centers of Masses, Moments of Inertia in Two Dimensions.
* Finding Masses, Moments, Centers of Masses, Moments of Inertia in Three Dimensions.
|-
|Week 13/14

||
5.7
||

[[Multiple Integrals|Change of Variables in Multiple Integrals]]

||

* [[Multiple Integrals|Double Integral]]
* [[Multiple Integrals|Triple Integrals]]

||

* Determine the image of a region under a given transformation of variables.
* Compute the Jacobian of a given transformation.
* Evaluate a double integral using a change of variables.
* Evaluate a triple integral using a change of variables.
|-
|Week 14

||
6.1
||

[[Vector Fields]]

||

* [[The Dot Product]] 
* [[Directional Derivatives and Gradient Vectors]]
||
* Vector Fields in a plane.
* Vector Fields in Space.
* Potential Functions.
* Gradient Fields, Conservative Vector Fields.
* The Cross-Partial Test for Conservative Vector Fields.
* Determining Whether a Vector Field is conservative.
|-

|Week 14

||
6.2
||

[[Line Integrals]]

||

* [[Parametric Equations]]
* [[Curves in Space and Vector-Valued Functions]]
* [[Arc Length]]

||
* Line Integrals of functions a long a smooth curves in a planer or in space
* Line Integrals of of vector fields along an oriented curves in a plane or space..
* Properties of Vector Line Integrals.
* Evaluating Line Integrals.
* Applications of line integrals: Calculating Arc Length, the Mass of a wire, Work done by a force, Flux across a curve, Circulation of a force.
|-

|Week 14/15

||
6.3
||

[[Conservative Vector Fields]]

||

* [[Vector Fields and Line Integrals]]
* [[Partial Derivatives]]

||
* Describe simple and closed curves
* Define connected and simply connected regions.
* Explain how to test a vector field to determine whether it is conservative.
* Find a potential function for a conservative vector field.
* Use the Fundamental Theorem for Line Integrals to evaluate a line integral of a vector field.
|-

|Weeks 14/15

||
6.4
||

[[Green's Theorem]]

[[Stokes' Theorem]]

||

* [[Vector Fields]]
* [[Line Integrals]]
* [[Partial Derivatives]]
* [[The Dot Product]]
* [[Line Integrals|Path Independence]]
* [[Conservative Vector Fields]]

||

* Circulation form of Green's Theorem.
* Flux Form of Green’s Theorem.
* Applying Green's Theorem to find Work, Flux.

|}

MAT1193

2023-06-15T20:07:36Z

Benjamin.sencindiver: /* Calculus for the Biosciences */ Making Use of Bullets consistent

== Calculus for the Biosciences ==

[https://catalog.utsa.edu/search/?P=MAT%201193 MAT 1193 Calculus for the Biosciences]. (3-0) 3 Credit Hours. (TCCN = MATH 2313)

Prerequisite: [[MAT1093|MAT 1093]] or an equivalent course or satisfactory performance on a placement examination. An introduction to calculus is presented using discrete-time dynamical systems and differential equations to model fundamental processes important in biological and biomedical applications. Specific topics to be covered are limits, continuity, differentiation, antiderivatives, definite and indefinite integrals, the fundamental theorem of calculus, differential equations, and the phase-plane. (Formerly MAT 1194. Credit can be earned for only one of the following: MAT 1193, MAT 1194, or MAT 1214.) May apply toward the Core Curriculum requirement in Mathematics. Generally offered: Fall, Spring, Summer. Course Fees: DL01 $72; LRC1 $12; LRS1 $45; STSI $21.

{| class="wikitable"
|-
! Date !! Topic !! Pre-requisite !! Student Learning Outcome
|-
| Week 1 || Review of [[Functions]] and [[Slope|Change]]
||
* [[Basic graphing skills]]
* [[The idea of a function]]
* Graphs of elementary functions (lines, parabola)
* Understanding of slope
* Periodic functions
||
* Define a function and connect to a real-world dynamical model
* Identify the parts of linear functions (slope, y-intercept).
* Demonstrate how to manipulate fractions.
* Identify power functions and polynomials.
* Identify exponential functions and their graphs in terms of exponential growth/decay.
* Identify logarithmic functions, graph and solve equations with log properties.
* Analyze graphs of the sine and cosine by recognizing amplitude and period.
* Identify and compute composite functions.
|-
| Week 2 || [[Derivatives Rates of Change|Instantaneous Rate of Change]]
||
* Evaluating functions
* Tangent lines
* Average rate of change
* Equations of a line (slope-intercept, point-slope)
||
* Compute the average rate of change (ARC).
* Compute the instantaneous rate of change (IRC)
* Comparing and contrasting ARC with IRC
* Defining velocity using the idea of a limit
* Visualizing the limit with tangent lines
|-
| Week 3 ||
* [[The Limit Laws]]
* [[The Limit of a Function]]
||
'''Example'''
||
* Recognize graphs of derivatives from original function
* Estimate the derivative of a function given table data and graphically
* Interpret the derivative with units and alternative notations (Leibniz)
* Use derivative to estimate value of a function
* Use the limit definition to define the derivative at a particular point and to define the derivative function
* Distinguish the definition of continuity of a function
* Apply derivatives to biological functions
|-
| Week 4 & 5 || [[Derivative Formulas]]
||
* Equations of lines
* [[Limits]]
* [[Composite functions]]
* [[Exponential]]
* [[Logarithmic]]
* [[Trigonometric]]
* [[Applications of Derivatives|Applications]]
||
* Use constant formula and power formula to differentiate functions along with the sum and difference rule
* Use differentiation to find the equation of a tangent line to make predictions using tangent line approximation
* Differentiate exponential and logarithmic functions
* Differentiate composite functions using the chain rule
* Differentiate products and quotients
* Differentiate trigonometric functions
* Applications of trigonometric function derivatives
|-
| Week 6 || [[Applications of Derivatives]]
||
* [[Maxima and Minima|Local & Global Maxima & Minima]]
* Concavity
||
* Detecting a local maximum or minimum from graph and function values
* Test for both local and global maxima and minima using first derivative test (finding critical points)
* Test for both local and global maxima and minima using second derivative test (testing concavity)
* Using concavity for finding inflection points
* Apply max and min techniques in real world applications in the field of Biology (logistic growth)
|-
| Week 7 || [[Accumulated Change]]
||
* Distance formula
* Summation formulas
||
* Approximate total change from rate of change
* Computing area with Riemann Sums
* Apply concepts of finding total change with Riemann Sums
|-
| Week 7 || [[The Definite Integral]]
||
* Summation formulas
||
* Approximate total change from rate of change
* Computing area with Riemann Sums
* Apply concepts of finding total change with Riemann Sums

|-
| Week 8 || [[Antiderivatives]] ||
* Basics in graphing
||
* Use the limit formula to compute a definite integral
* Be able to analyze area under the curve with antiderivatives graphically and numerically
* Use formulas for finding antiderivatives of constants and powers
* Use formulas for finding antiderivatives of exponential and logarithm functions
* Use formulas for finding antiderivatives of trigonometric functions
|-
| Week 9 || [[The Fundamental Theorem of Calculus]] ||
*Average formula
||
* Use the limit formula to compute a definite integral
* Compute area with the fundamental theorem of calculus (FTC)

|-
| Week 10 || [[Applications of Integrals]] || '''Example''' ||
* Solve various biology applications using the fundamental theorem of calculus
|-
| Week 11|| [[Integration by Substitution]] || '''Example''' ||
* Applying integration by substitution formulas
|-
| Week 12|| [[Integration by Parts]] || '''Example'''
||
* Applying integration by integration by parts formulas
* Recognize which integration formulas to use
|-
| Week 13|| [[Differential Equations (Mathematical Modeling)]] ||
* Word problem setup and understanding of mathematical models
||
* Demonstrate how to take information to set up a mathematical model
* Examine the basic parts of differential equations
|-
| Week 14|| [[Differential Equations]] ||
* Graphing and factoring
||
* Examine differential equations graphically with slope fields
* Use separation of variables for solving differential equations
|-
| Week 15|| [[Differential Equations (Mathematical Modeling)|Differential Equations Applications]] ||
* [[Exponential functions]]
||
* Apply differential equations to exponential growth & decay functions for population models
* Apply differential equations to surge functions for drug models
* Modeling the spread of a disease
|}

Talk:MAT2214

2023-06-15T19:53:24Z

Benjamin.sencindiver: Suggestion for separating row by topics (for Section 6.4)

* In the initial plan for this course, the first topics are
Three-Dimensional Coordinate Systems, vectors, dot product, cross product; these topics are covered in Linear Algebra. Conics could be moved to Precalculus. Instead a brief review session could be added. This opens slots for other topics, or emphasis in certain area (JK - 7/3/2020).

* Ben Sencindiver - 6/15/2023 - Final section (6.4) has two topics in the final row. I suggest separating these into two rows (each with same 6.4 section associated). This involves separating the prerequisite skills and student learning outcomes as well.

MAT2214

2023-06-15T19:49:42Z

Benjamin.sencindiver: /* Topics List */ Minor edits to fix font on Section Numbers

The textbook for this course is
[https://openstax.org/details/books/calculus-volume-3 Calculus (Volume 3) by Gilbert Strang, Edwin Herman, et al.]

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-

|Week 1

||
1.1
||

[[Polar Coordinates]]

||
* [[Trigonometric Functions: Unit Circle Approach]]
* [[Inverse Trigonometric Functions]]
||
* Plot points using polar coordinates and find several polar coordinates of a single point
* Convert polar coordinates to rectangular coordinates and vice versa
* Transform equations from polar form to rectangular form and vice versa
|-

|Week 1

||
1.2
||

[[Three-Dimensional Coordinate Systems]]

||

* [[Two-dimensional coordinate systems]]
* [[Solving Equations and Inequalities| Algebraic Expressions]]

||

* Three-dimensional coordinate systems.
* Distance Formula in Space.
* Standard Equation for a Sphere.
|-

|Weeks 1/2

||
2.1
||

[[Vectors in The Plane, Space]]

||

* [[Linear Equations|Line Segments]]
* [[Distance Formula| Distance Formula]]

||

* Vector Algebra Operations
* The Magnitude of a vector
* Unit Vectors
* The Midpoint of a Line Segment
* The Vector projection
|-

|Week 2

||
2.3
||

[[The Dot Product]]

||

* [[Trigonometric Functions|Basic Trig Functions]]
* [[Vectors]]

||
* Definition of Dot Product
* Properties of Dot Product
* Angle between vectors
* Orthogonal vectors

|-

|Week 2

||
2.4
||

[[The Cross Product]]

||

* [[Trigonometric Functions|Basic Trig Functions]]
* [[Determinants]]
* [[Vectors]]

||

* Definition of Cross Product
* Properties of the cross product
* Area of a parallelogram
* Cross product as a determinant

|-

|Week 3

||
2.5
||

[[Equations of Lines, Planes and Surfaces in Space]]

||

* [[The Dot Product]]
* [[The Cross Product]]
* [[Quadratic Functions]]
* [[Parametric Equations]]

||

* Write the vector, parametric equation of a line through a given point in a given direction, and a line through two given points.
* Find the distance from a point to a given line.
* Write the equation of a plane through a given point with a given normal, and a plane through three given points.
* Find the distance from a point to a given plane.

|-

|Week 3

||
2.6
||

[[Equations of Lines, Planes and Surfaces in Space|Cylinders and Quadratic Surfaces]]

||

* [[Quadratic Functions]]
* [[Parametric Equations]]
* '''[[Conics]]'''

||

* Find equations for cylinders that are generated by rotating lines that are parallel to a plane
* Understand basic quadratic surfaces
* Understand general quadratic surfaces

|-

|Weeks 3/4

||
3.1, 3.2
||

[[Curves in Space and Vector-Valued Functions]]

||

* [[Parametric Equations]]
* [[Vectors]]
* [[The Derivative as a Function]]
* [[The Limit of a Function]]
* [[Continuity]]
* [[The Dot Product]]
* [[The Cross Product]]

||

* Vector functions
* Limits of vector functions
* Continuity of vector functions
* Differentiation rules for vector functions
* Curves and paths in space

|-

|Week 4

||
3.3
||

[[Arc Length]]

||

* '''[[Distance Formula| The Length of a Line Segment]]'''
* [[Curves in Space and Vector-Valued Functions|Vector Functions]]
* [[Line Integrals|Integrals of Vector Functions]], [[Derivatives of Vector Functions]]

||

* The arc Length of a vector function
* Arc length parameterization

|-

|Weeks 4/5

||
3.4
||

[[Motion in Space]]

||
* [[Vectors]]
* [[Parametric Equations]]
* [[The Cross Product]]
* [[Derivatives of Vector Functions]]
||
* The Unit tangent vector
* The curvature
* The Principal Unit Normal Vector
* The Binormal Vector
* The tangential and normal components of acceleration
* The Torsion

|-

|Week 5/6

||
4.1
||

[[Functions of Several Variables]]

||

* [[Domain of a Function]]
* [[Range of a Function]]
* [[Solving Equations and Inequalities]]
* [[Graphs| Graphing a Function]]

||
* Functions of two variables
* Functions of three variables
* Domain and range of multivariable functions
* Bounded regions
* Graphs and level curves of two variable functions
* Level surfaces of three variable functions
|-

|Week 6

||
4.2
||

[[Limit and Continuity of Function of Several Variables]]

||

* [[Continuity]]
* [[The Limit Laws]]
* [[Composition of Functions]]
* [[The Dot Product]]

||

* Limits of functions of two variables
* Limits of functions of more than two variables
* Properties of limits of functions of several variables
* Two path test of non-existing of a limit
* Continuity for functions of several variables
* Continuity of composition
* Extreme values on closed and bounded domains
|-

|Week 6

||
4.3
||

[[Partial Derivatives]]

||

* [[The Derivative as a Function|The first]] and [[The Second Derivative|second derivative]] of a function
* [[Limit and Continuity of Function of Several Variables]]

||
* Partial derivatives for functions of two variables
* Partial derivatives for functions of more than two variables
* Partial derivatives and continuity
* Second order partial derivatives
* Mixed derivative theorem
|-

|Week 7

||
4.4
||

[[Directional Derivatives and Gradient Vectors]]

||
* [[Trigonometric Functions]]
* [[Vectors, Unit Vectors]]
* [[Partial Derivatives]]
* [[Gradients]]
* [[The Dot Product]]
||
* Directional derivatives for functions of two variables
* Gradients
* Properties of directional derivatives
* Tangents to level curves
* Directional derivatives for functions of three variables
|-

|Week 7

||
4.5
||
[[Tangent Plane]],
[[Differentiability]]

||
* [[Partial Derivatives]]
* [[Parametric Equations]] of Lines
* [[Equations of Lines, Planes and Surfaces in Space]]

||
* Determine the equation of a plane tangent to a given surface at a point
* Determine the parametric equation of a normal line to a given surface at a point
* The linear approximation of a function of two variables at a point
* The definition of differentiability for a function of two variables
* Differentiability implies Continuity
* Continuity of First Partial Derivatives implies Differentiability
* The definition of total differentiability for a function of two variables
* Use the total differential to approximate the change in a function of two variables
|-

|Week 7

||
4.6
||

[[The Chain Rule for Functions of more than One Variable]]

||
* [[Differentiation Rules]]
* [[The Chain Rule]]
* [[Partial Derivatives]]

||
* Chain rule for functions of one independent variable and several intermediate variables.
* Chain rule for functions of two independent variable and several intermediate variables.
* Method for implicit differentiation.
* The general chain rule for functions of several independent variables
|-
|Week 8

||
4.7
||

[[Maxima and Minima Problems]]

||
* [[Extreme values on closed and bounded domains]]
* [[Partial Derivatives]]
* [[Maxima and Minima|Maxima, Minima and Critical Points of a Function]]
* [[Limit and Continuity of Function of Several Variables]]

||
* The derivative test for local extreme values
* Extreme values on closed and bounded domains
* Critical points and saddle points for functions of two variables
* Second derivative test for local extreme values
* Absolute maxima and minima on closed and bounded regions
|-
|Week 8/9

||
4.8
||

[[Lagrange Multipliers]]

||

* [[Partial Derivatives]]
* [[Critical Points of a Function]]

||
* Lagrange Multipliers with One Constraint
* Lagrange Multipliers with Two Constraints
|-

|Week 9/10

||
5.1
||

[[Multiple Integrals|Double Integrals over Rectangular Regions]]

||

* [[Approximating Areas]]
* [[The Definite Integral|Limits of Riemann Sums]]

||
* Double Integral is the limit of Double Sums.
* Double Integrals over Rectangular Regions.
* Interated Integrals.
* Fubini's Theorem (part 1).
|-

|Week 10

||
5.2
||

[[Multiple Integrals|Double Integrals over General Regions]]

||

* [[Continuity]]
* [[Determining Volumes by Slicing]]
* [[Multiple Integrals|Double and Iterated Integrals over Rectangular regions]]

||
* Double integrals over bounded, general regions.
* Properties of double Integrals.
* Fubini's theorem (part 2)
* Changing the order of Integration.
* Calculating Volumes, Areas and Average Values
|-

|Week 11

||
5.3
||

[[Multiple Integrals|Double Integrals in Polar Coordinates]]

||

* [[Multiple Integrals|Double Integrals over General Regions]]
* [[Polar Coordinates]]

||
* Double Integrals over rectangular polar regions.
* Double Integrals over general polar regions.
* Changing Cartesian Integrals into Polar Integrals.
* Using Double Integrals in Polar Coordinates to find Volumes, Areas.
|-

|Week 11

||
5.4
||

[[Multiple Integrals|Triple Integrals in Rectangular Coordinates]]

||

* [[Multiple Integrals|Double Integrals]]
* [[Multiple Integrals|Area by Double Integration]]
* '''[[Change of Variables]]'''

||

* Triple Integrals over general bounded regions.
* Finding Volumes by evaluating Triple Integrals.
* Average value of a function in space.
* Changing Integration Order and Coordinate systems.
|-

|Week 12

||
5.5
||

[[Multiple Integrals|Triple Integrals in Cylindrical and Spherical Coordinates]]

||

* [[Multiple Integrals|Double Integrals in Polar Form]]
* [[Multiple Integrals|Triple Integrals in Rectangular Coordinates]]

||

* Integrations in Cylindrical Coordinates.
* Equations relating rectangular and cylindrical coordinates.
* Changing Cartesian integrations into Cylindrical integrations.
* Integrations in Spherical coordinates.
* Equations relating spherical coordinates to Cartesian and cylindrical coordinates.
* Changing Cartesian integrations into Cylindrical integrations.
|-

|Week 13

||
5.6
||

[[Multiple Integrals|Applications of Multiple Integrals]]

||

* [[Multiple Integrals|Double Integral]]
* [[Multiple Integrals|Triple Integrals]]

||

* Finding Masses, Moments, Centers of Masses, Moments of Inertia in Two Dimensions.
* Finding Masses, Moments, Centers of Masses, Moments of Inertia in Three Dimensions.
|-
|Week 13/14

||
5.7
||

[[Multiple Integrals|Change of Variables in Multiple Integrals]]

||

* [[Multiple Integrals|Double Integral]]
* [[Multiple Integrals|Triple Integrals]]

||

* Determine the image of a region under a given transformation of variables.
* Compute the Jacobian of a given transformation.
* Evaluate a double integral using a change of variables.
* Evaluate a triple integral using a change of variables.
|-
|Week 14

||
6.1
||

[[Vector Fields]]

||

* [[The Dot Product]] 
* [[Directional Derivatives and Gradient Vectors]]
||
* Vector Fields in a plane.
* Vector Fields in Space.
* Potential Functions.
* Gradient Fields, Conservative Vector Fields.
* The Cross-Partial Test for Conservative Vector Fields.
* Determining Whether a Vector Field is conservative.
|-

|Week 14

||
6.2
||

[[Line Integrals]]

||

* [[Parametric Equations]]
* [[Curves in Space and Vector-Valued Functions]]
* [[Arc Length]]

||
* Line Integrals of functions a long a smooth curves in a planer or in space
* Line Integrals of of vector fields along an oriented curves in a plane or space..
* Properties of Vector Line Integrals.
* Evaluating Line Integrals.
* Applications of line integrals: Calculating Arc Length, the Mass of a wire, Work done by a force, Flux across a curve, Circulation of a force.
|-

|Week 14/15

||
6.3
||

[[Conservative Vector Fields]]

||

* [[Vector Fields and Line Integrals]]
* [[Partial Derivatives]]

||
* Describe simple and closed curves
* Define connected and simply connected regions.
* Explain how to test a vector field to determine whether it is conservative.
* Find a potential function for a conservative vector field.
* Use the Fundamental Theorem for Line Integrals to evaluate a line integral of a vector field.
|-

|Weeks 14/15

||
6.4
||

[[Green's Theorem]]

[[Stokes' Theorem]]

||

* [[Vector Fields]]
* [[Line Integrals]]
* [[Partial Derivatives]]
* [[The Dot Product]]
* [[Line Integrals|Path Independence]]
* [[Conservative Vector Fields]]

||

* Circulation form of Green's Theorem.
* Flux Form of Green’s Theorem.
* Applying Green's Theorem to find Work, Flux.

|}

Talk:The Limit of a Function

2023-06-15T17:23:11Z

Benjamin.sencindiver: Suggestion for removing content.

* Ben S - 6/15/2023 - Limits of summations and integrals is not appropriate students learning about limits of a function for the first time (1193, Week 3). Recommend removing definite integral content.

Talk:MAT1093

2023-06-15T16:57:26Z

Benjamin.sencindiver: Added Suggestion for Removing prereq in 1.3

* In the initial plan, weeks 13, 14 are devoted to logarithmic functions, Properties of logarithms, Log and exp equations, Exp. growth and decay models, Newton’s law of Cooling, Logistic growth and decay. These topics are covered in College Algebra (MAT1073), and shoudl be removed from this course (JBG)

* Conics should be introduced here because they are needed in Calc III (JBG)

* Week 1 Discrepancy between Functions from MAT 1073 and Functions and their Graphs from MAT 1093, suggestion to edit so that they match.

* Rene G - 6/15/2023 - "Evaluating Algebraic Expressions" Prereq might not be necessary for section 1.3. Lesson does not involve evaluating any algebraic expressions. Suggest to remove this prereq.