4.1 Tangent planes to surfaces

Tangent Planes and Normal Vectors

A tangent plane is the flat surface that best approximates a smooth surface at a given point. If you zoom in far enough on a smooth surface near that point, the surface and its tangent plane become nearly indistinguishable. Understanding tangent planes is essential for linear approximation in multiple variables and for analyzing how surfaces behave locally.

Finding tangent planes relies on partial derivatives and gradient vectors. This section covers how to compute the equation of a tangent plane, how gradients connect to normal vectors, and how level surfaces tie everything together.

Defining Tangent Planes and Normal Vectors

A tangent plane to a surface at a point $(x_0, y_0, z_0)$ is the plane that just touches the surface at that point and matches the surface's "tilt" in every direction. For a smooth surface, the tangent plane contains all the tangent lines to curves on the surface passing through that point.

A normal vector $\vec{n} = \langle a, b, c \rangle$ is a vector perpendicular to the tangent plane at the point of tangency. Once you know a normal vector and the point, the tangent plane is fully determined.

• The normal vector is computed using partial derivatives of the surface equation (details below).
• Every nonzero scalar multiple of $\vec{n}$ is also a valid normal vector, so the normal direction is unique but the vector itself is not.

Finding the Equation of a Tangent Plane

The general equation of a tangent plane at $(x_0, y_0, z_0)$ with normal vector $\langle a, b, c \rangle$ is:

$a(x - x_0) + b(y - y_0) + c(z - z_0) = 0$

Case 1: Surface given as $z = f(x, y)$

Rewrite the surface as $F(x, y, z) = f(x, y) - z = 0$. The normal vector at $(x_0, y_0, z_0)$ is then $\langle f_x(x_0, y_0),\; f_y(x_0, y_0),\; -1 \rangle$, and the tangent plane equation becomes:

$z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)$

Step-by-step process:

1. Compute $f_x$ and $f_y$.
2. Evaluate both at the point $(x_0, y_0)$.
3. Plug into the formula above, using $z_0 = f(x_0, y_0)$.

Example: Find the tangent plane to $z = x^2 + y^2$ at $(1, 1, 2)$.

1. $f_x = 2x$, $f_y = 2y$.
2. At $(1,1)$: $f_x = 2$, $f_y = 2$.
3. Tangent plane: $z - 2 = 2(x - 1) + 2(y - 1)$, which simplifies to $2(x-1) + 2(y-1) - (z-2) = 0$.

The normal vector here is $\langle 2, 2, -1 \rangle$.

Case 2: Surface given implicitly as $F(x, y, z) = c$

Use the gradient $\nabla F$ evaluated at the point as the normal vector. The tangent plane equation is:

$F_x(x_0,y_0,z_0)(x - x_0) + F_y(x_0,y_0,z_0)(y - y_0) + F_z(x_0,y_0,z_0)(z - z_0) = 0$

This is the more general approach and works for any smooth surface, including level surfaces.

Partial Derivatives and Gradients

Partial Derivatives in Multivariable Functions

Partial derivatives measure the rate of change of a function with respect to one variable while all other variables are held constant.

• $\frac{\partial f}{\partial x}$ is computed by differentiating $f$ with respect to $x$, treating $y, z, \ldots$ as constants.
• $\frac{\partial f}{\partial y}$ does the same but with respect to $y$.

For tangent planes, partial derivatives tell you the slope of the surface in the $x$- and $y$-directions independently. Together they determine how the surface tilts at a point.

Gradient Vector and Its Applications

The gradient vector of a function $f(x, y, z)$ is:

$\nabla f = \left\langle \frac{\partial f}{\partial x},\; \frac{\partial f}{\partial y},\; \frac{\partial f}{\partial z} \right\rangle$

Two key properties make the gradient central to tangent plane problems:

• $\nabla f$ points in the direction of greatest rate of increase of $f$.
• $\nabla f$ is perpendicular to the level surface $f(x,y,z) = c$ at every point on that surface. This means the gradient is the normal vector you need for the tangent plane.

Example: For $F(x, y, z) = x^2 + y^2 - z$, the gradient is $\nabla F = \langle 2x, 2y, -1 \rangle$. At $(1, 1, 2)$, this gives $\nabla F = \langle 2, 2, -1 \rangle$, which serves as the normal vector to the tangent plane.

Level Surfaces

Understanding Level Surfaces

A level surface of a function $f(x, y, z)$ is the set of all points where $f$ takes a constant value:

$f(x, y, z) = c$

These are the 3D analog of contour lines (level curves) for functions of two variables. Each value of $c$ gives a different surface.

Example: For $f(x, y, z) = x^2 + y^2 + z^2$, the level surfaces are spheres centered at the origin. The level surface $f = 4$ is a sphere of radius 2; $f = 9$ is a sphere of radius 3.

Relationship between Level Surfaces and Gradient Vectors

The gradient $\nabla f$ at any point on a level surface $f(x,y,z) = c$ is perpendicular to that surface. This is the fundamental link between gradients and tangent planes:

• To find the tangent plane to a level surface at a point, compute $\nabla f$ at that point and use it as the normal vector.
• This works because any curve lying on the level surface has a tangent vector perpendicular to $\nabla f$, so $\nabla f$ must be normal to the surface.

Example: For $f(x,y,z) = x^2 + y^2 + z^2$ at the point $(1, 1, 1)$ on the level surface $f = 3$:

1. $\nabla f = \langle 2x, 2y, 2z \rangle$. At $(1,1,1)$: $\nabla f = \langle 2, 2, 2 \rangle$.
2. Tangent plane: $2(x-1) + 2(y-1) + 2(z-1) = 0$, which simplifies to $x + y + z = 3$.

The gradient points radially outward from the origin, which makes geometric sense for a sphere.