Tangent Plane

Intuitively, it seems clear that, in a plane, only one line can be tangent to a curve at a point. However, in three-dimensional space, many lines can be tangent to a given point. If these lines lie in the same plane, they determine the tangent plane at that point. A tangent plane at a regular point contains all of the lines tangent to that point. A more intuitive way to think of a tangent plane is to assume the surface is smooth at that point (no corners). Then, a tangent line to the surface at that point in any direction does not have any abrupt changes in slope because the direction changes smoothly.

Definition of Tangent Plane

Tangent plane

Let

P_{0}=(x_{0},y_{0},z_{0})

be a point on a surface

S

, and let

C

be any curve passing through

P_{0}

and lying entirely in

S

. If the tangent lines to all such curves

C

at

P_{0}

lie in the same plane, then this plane is called the tangent plane to

S

at

P_{0}

.

For a tangent plane to a surface to exist at a point on that surface, it is sufficient for the function that defines the surface to be differentiable at that point, defined later in this section. We define the term tangent plane here and then explore the idea intuitively.

Equation of Tangent Plane

Let

S

be a surface defined by a differentiable function

z=f(x,y)

, and let

P_{0}=(x_{0},y_{0})

be a point in the domain of

f

. Then, the equation of the tangent plane to

S

at

P_{0}

is given by

z=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})

To see why this formula is correct, let’s first find two tangent lines to the surface S. The equation of the tangent line to the curve that is represented by the intersection of S with the vertical trace given by $x=x_{0}$ is $z=f(x_{0},y_{0})+f_{y}(x_{0},y_{0})(y-y_{0})$ . Similarly, the equation of the tangent line to the curve that is represented by the intersection of $S$ with the vertical trace given by $y=y_{0}$ is $z=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})$ . A parallel vector to the first tangent line is $a=\mathbf {j} +f_{y}(x_{0},y_{0})\mathbf {k}$ ; a parallel vector to the second tangent line is $b=\mathbf {i} +f_{x}(x_{0},y_{0})\mathbf {k}$ . We can take the cross product of these two vectors:

a\times b=(\mathbf {j} +f_{y}(x_{0},y_{0})\mathbf {k} )\times (\mathbf {i} +f_{x}(x_{0},y_{0})\mathbf {k} )

=f_{x}(x_{0},y_{0})\mathbf {i} +f_{y}(x_{0},y_{0})\mathbf {j} -\mathbf {k}

This vector is perpendicular to both lines and is therefore perpendicular to the tangent plane. We can use this vector as a normal vector to the tangent plane, along with the point $P_{0}=(x_{0},y_{0},f(x_{0},y_{0}))$ in the equation for a plane:

n\cdot ((x-x_{0})\mathbf {i} +(y-y_{0})\mathbf {j} +(z-f(x_{0},y_{0}))\mathbf {k} )=0

\implies (f_{x}(x_{0},y_{0})\mathbf {i} +f_{y}(x_{0},y_{0})\mathbf {j} -\mathbf {k} )\cdot ((x-x_{0})i+(y-y_{0})\mathbf {j} +(z-f(x_{0},y_{0}))\mathbf {k} )=0

\implies f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})-(z-f(x_{0},y_{0}))=0

\implies z=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})

Linear Approximations

When working with a function of two variables, the tangent line is replaced by a tangent plane, but the approximation idea is much the same.

Definition of Linear Approximation Given a function $z=f(x,y)$ with continuous partial derivatives that exist at the point $(x_{0},y_{0})$ , the linear approximation of $f$ at the point $(x_{0},y_{0})$ is given by the equation

L(x,y)=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})

Notice that this equation also represents the tangent plane to the surface defined by $z=f(x,y)$ at the point $(x_{0},y_{0})$ . The idea behind using a linear approximation is that, if there is a point $(x_{0},y_{0})$ at which the precise value of $f(x,y)$ is known, then for values of $(x,y)$ reasonably close to $(x0,y0)$ , the linear approximation (i.e., tangent plane) yields a value that is also reasonably close to the exact value of $f(x,y)$ . Furthermore the plane that is used to find the linear approximation is also the tangent plane to the surface at the point $(x_{0},y_{0})$ .