22.2 Tangent planes and normal vectors to surfaces

Tangent Planes and Normal Vectors

Defining Tangent Planes and Normal Vectors

A tangent plane is the plane that best approximates a surface at a given point. Think of it as the flat surface you'd feel if you pressed your palm against a curved object at that spot. Every direction you could "slide" along the surface at that point lies within the tangent plane.

A normal vector ($\vec{n}$ or $\vec{N}$) is a vector perpendicular to the tangent plane at the point of tangency. It sticks straight out from the surface. You can find it by taking the cross product of two independent tangent vectors at that point.

Partial derivatives are the key tool here. For a function $f(x, y, z)$, the partial derivatives $f_x$, $f_y$, and $f_z$ tell you how $f$ changes as you move along each coordinate direction. These partials are what let you build both tangent planes and normal vectors.

Finding Tangent Planes and Normal Vectors

There are two main setups you'll encounter, depending on how the surface is defined.

Case 1: Implicit surfaces $f(x, y, z) = c$

The gradient vector $\nabla f(x, y, z) = \langle f_x, f_y, f_z \rangle$ points in the direction of greatest rate of change of $f$. A critical fact: the gradient is always perpendicular to the level surface $f = c$ at any point. That means $\nabla f$ evaluated at your point is the normal vector.

To find the tangent plane to $f(x, y, z) = c$ at a point $(x_0, y_0, z_0)$:

1. Compute the gradient $\nabla f(x_0, y_0, z_0) = \langle f_x, f_y, f_z \rangle$ evaluated at the point.
2. Write the tangent plane using point-normal form:

$f_x(x_0, y_0, z_0)(x - x_0) + f_y(x_0, y_0, z_0)(y - y_0) + f_z(x_0, y_0, z_0)(z - z_0) = 0$

This is equivalent to $\nabla f(x_0, y_0, z_0) \cdot \langle x - x_0, y - y_0, z - z_0 \rangle = 0$.

Case 2: Parametric surfaces $\vec{r}(u, v) = \langle x(u,v),\, y(u,v),\, z(u,v) \rangle$

Here you don't have a gradient to work with directly. Instead, you build the normal from the two coordinate tangent vectors.

1. Compute $\vec{r}_u(u_0, v_0)$ and $\vec{r}_v(u_0, v_0)$ by taking partial derivatives of $\vec{r}$ with respect to $u$ and $v$, then evaluating at the point.
2. The normal vector is their cross product: $\vec{n} = \vec{r}_u(u_0, v_0) \times \vec{r}_v(u_0, v_0)$.
3. The tangent plane passes through $\vec{r}(u_0, v_0)$ with normal $\vec{n}$.

Note that $\vec{r}_u$ and $\vec{r}_v$ must be linearly independent (not parallel) at the point for this to work. If they're parallel, the surface has a singularity there and the tangent plane isn't well-defined.

Directional Derivatives and Tangent Vectors

Understanding Directional Derivatives

Partial derivatives only measure change along the coordinate axes. A directional derivative generalizes this: it measures the rate of change of $f$ in any direction you choose.

For a unit vector $\vec{u}$, the directional derivative is:

$D_{\vec{u}}f(x, y, z) = \nabla f(x, y, z) \cdot \vec{u}$

Two important consequences follow from this dot product formula:

• $D_{\vec{u}}f$ is maximized when $\vec{u}$ points in the same direction as $\nabla f$. The maximum value equals $\|\nabla f\|$.
• $D_{\vec{u}}f = 0$ when $\vec{u}$ is perpendicular to $\nabla f$. This confirms that moving along the level surface (tangent to it) produces no change in $f$.

Tangent Vectors and the Cross Product

For a parametric surface $\vec{r}(u, v)$, the vectors $\vec{r}_u(u_0, v_0)$ and $\vec{r}_v(u_0, v_0)$ are tangent vectors at the point. They span the tangent plane, meaning every tangent direction at that point can be written as a linear combination of these two vectors.

The cross product $\vec{a} \times \vec{b}$ produces a vector perpendicular to both $\vec{a}$ and $\vec{b}$. Two properties matter here:

• The magnitude $\|\vec{a} \times \vec{b}\|$ equals the area of the parallelogram spanned by $\vec{a}$ and $\vec{b}$. This connects directly to surface area calculations in the next sections of this unit.
• The direction is determined by the right-hand rule: curl your fingers from $\vec{a}$ toward $\vec{b}$, and your thumb points in the direction of $\vec{a} \times \vec{b}$.

So taking $\vec{r}_u \times \vec{r}_v$ gives you a vector perpendicular to both tangent directions, which is exactly the normal vector to the surface.

Parametric Curves on Surfaces

Defining Parametric Curves on Surfaces

A parametric curve on a surface is a path that stays on the surface as a single parameter $t$ varies. If the surface is $\vec{r}(u, v)$, you get such a curve by letting $u$ and $v$ each depend on $t$:

$\vec{r}(u(t), v(t))$

The tangent vector to this curve comes from the chain rule:

$\frac{d\vec{r}}{dt} = \vec{r}_u \frac{du}{dt} + \vec{r}_v \frac{dv}{dt}$

Notice this tangent vector is a linear combination of $\vec{r}_u$ and $\vec{r}_v$, which means it lies in the tangent plane. That makes geometric sense: a curve confined to the surface can only move in directions tangent to the surface.

The normal vector to the surface at a point on the curve is still $\vec{r}_u \times \vec{r}_v$ evaluated at that point. The curve doesn't change the surface's normal; it just picks out which points on the surface you're visiting.

Examples of Parametric Curves on Surfaces

A circle on a sphere. The unit sphere can be parametrized as $\vec{r}(u, v) = \langle \cos u \cos v,\, \sin u \cos v,\, \sin v \rangle$. Setting $v = 0$ and letting $u = t$ gives the equator:

$\vec{r}(t) = \langle \cos t,\, \sin t,\, 0 \rangle$

This traces a great circle in the $xy$-plane, and it stays on the sphere since $\cos^2 t + \sin^2 t + 0^2 = 1$.

A helix on a cylinder. The cylinder $x^2 + y^2 = 1$ can be parametrized as $\vec{r}(u, v) = \langle \cos u,\, \sin u,\, v \rangle$. Setting $u = t$ and $v = t$ gives:

$\vec{r}(t) = \langle \cos t,\, \sin t,\, t \rangle$

This spirals upward along the cylinder, winding around it while climbing at a constant rate.