Calculus 3 Review

This review covers everything from Calculus 3: vectors in 3D space, vector functions, partial derivatives, multiple integrals, vector fields, line and surface integrals, and the three big theorems (Green’s, Stokes’, Divergence). Use this as your final checkpoint.

The dot product $\mathbf{A} \cdot \mathbf{B} = \|\mathbf{A}\|\|\mathbf{B}\|\cos\theta$ measures how aligned two vectors are. The cross product $\mathbf{A} \times \mathbf{B}$ gives a vector perpendicular to both, with magnitude equal to the area of the parallelogram they form. Lines in 3D are parametrized as $\mathbf{r}(t) = \mathbf{r}_0 + t\mathbf{v}$, and planes are described by $\mathbf{n} \cdot (\mathbf{r} - \mathbf{r}_0) = 0$.

A space curve is parametrized by $\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle$. The tangent vector is $\mathbf{r}'(t)$, the unit tangent is $\mathbf{T} = \mathbf{r}'/\|\mathbf{r}'\|$, and arc length is $L = \int_a^b \|\mathbf{r}'(t)\| \, dt$. Curvature $\kappa$ measures how fast the curve turns.

For $f(x,y,z)$, partial derivatives $f_x, f_y, f_z$ measure the rate of change in each direction independently. The gradient $\nabla f = \langle f_x, f_y, f_z \rangle$ points in the direction of steepest ascent. The directional derivative in direction $\mathbf{u}$ is $D_\mathbf{u}f = \nabla f \cdot \mathbf{u}$. Tangent planes and linear approximations extend the idea of tangent lines to surfaces.

Double integrals compute area, volume, and accumulated quantities over 2D regions. Triple integrals extend this to 3D. The key is choosing the right coordinate system:

• Rectangular: $dA = dx\,dy$, $dV = dx\,dy\,dz$
• Polar: $dA = r\,dr\,d\theta$
• Cylindrical: $dV = r\,dr\,d\theta\,dz$
• Spherical: $dV = \rho^2 \sin\varphi\,d\rho\,d\varphi\,d\theta$

The extra factors (r, $\rho^2\sin\varphi$) are Jacobians that account for how the coordinate grid stretches.

A vector field assigns a vector to each point: $\mathbf{F}(x,y,z) = \langle P, Q, R \rangle$. The line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$ computes work done by the field along a path. The surface integral $\iint_S \mathbf{F} \cdot d\mathbf{S}$ computes flux through a surface. Conservative fields have a potential function $f$ where $\mathbf{F} = \nabla f$, and their line integrals are path-independent.

These theorems connect integrals across different dimensions. They all say the same thing: what happens on the boundary determines what’s happening inside.

Green’s Theorem (2D): line integral around a closed curve = double integral over the enclosed region

$$\oint_C P\,dx + Q\,dy = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA$$

Stokes’ Theorem (3D surfaces): surface integral of curl = line integral around the boundary

$$\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \oint_C \mathbf{F} \cdot d\mathbf{r}$$

Divergence Theorem (3D volumes): flux through a closed surface = triple integral of divergence inside

$$\oiint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_E (\nabla \cdot \mathbf{F})\,dV$$

Curl measures rotation. Divergence measures sources and sinks. If curl F = 0 (and the domain is simply connected), F is conservative. If div F = 0, the field is incompressible.

Example 1: Divergence Theorem

Compute the outward flux of $\mathbf{F} = \langle x, y, z \rangle$ through the unit sphere.

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Instead of parametrizing the sphere, use the Divergence Theorem. The divergence is:

$$\nabla \cdot \mathbf{F} = 1 + 1 + 1 = 3$$

Integrate over the unit ball:

$$\oiint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_E 3\,dV = 3 \cdot \frac{4}{3}\pi = 4\pi$$

Example 2: Stokes’ Theorem

Evaluate the circulation of $\mathbf{F} = \langle -y, x, 0 \rangle$ around the unit circle in the xy-plane.

Compute the curl:

$$\nabla \times \mathbf{F} = \langle 0, 0, 2 \rangle$$

The flat unit disk has upward normal $\mathbf{n} = \langle 0,0,1 \rangle$, so:

$$\iint_D (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \iint_D 2\,dA = 2\pi$$

Example 3: Volume in Spherical Coordinates

Find the volume of the unit ball $x^2 + y^2 + z^2 \leq 1$.

In spherical coordinates, $\rho$ (rho) is the distance from the origin, $\varphi$ (phi) is the angle from the positive z-axis, and $\theta$ (theta) is the angle in the xy-plane. The volume element is $dV = \rho^2 \sin\varphi\,d\rho\,d\varphi\,d\theta$.

$$V = \int_0^{2\pi} \int_0^{\pi} \int_0^{1} \rho^2 \sin\varphi \, d\rho \, d\varphi \, d\theta$$

The innermost integral:

$$\int_0^1 \rho^2 \, d\rho = \frac{1}{3}$$

The middle integral:

$$\int_0^{\pi} \sin\varphi \, d\varphi = 2$$

The outer integral:

$$\int_0^{2\pi} d\theta = 2\pi$$

Multiply them together: $V = \frac{1}{3} \cdot 2 \cdot 2\pi = \frac{4\pi}{3}$.

Example 4: Line Integral (Work)

Compute the work done by $\mathbf{F} = \langle -y, x \rangle$ along the unit circle from (1,0) back to (1,0), counterclockwise.

Parametrize: $\mathbf{r}(t) = \langle \cos t, \sin t \rangle$ for $t$ from 0 to $2\pi$.

$$\mathbf{r}'(t) = \langle -\sin t, \cos t \rangle$$ $$\mathbf{F}(\mathbf{r}(t)) = \langle -\sin t, \cos t \rangle$$ $$\mathbf{F} \cdot \mathbf{r}' = (-\sin t)(-\sin t) + (\cos t)(\cos t) = \sin^2 t + \cos^2 t = 1$$ $$W = \int_0^{2\pi} 1\,dt = 2\pi$$

Example 5: Green’s Theorem

Evaluate $\oint_C (x^2 + y)\,dx + (3x - y)\,dy$ where C is the unit circle.

Using Green’s Theorem with $P = x^2 + y$ and $Q = 3x - y$:

$$\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 3 - 1 = 2$$ $$\oint_C P\,dx + Q\,dy = \iint_D 2\,dA = 2 \cdot \pi(1)^2 = 2\pi$$

Example 6: Curl of a Rotational Field

For $\mathbf{F} = \langle -y, x, 0 \rangle$, compute the curl.

$$\nabla \times \mathbf{F} = \left\langle \frac{\partial 0}{\partial y} - \frac{\partial x}{\partial z},\; \frac{\partial(-y)}{\partial z} - \frac{\partial 0}{\partial x},\; \frac{\partial x}{\partial x} - \frac{\partial(-y)}{\partial y} \right\rangle = \langle 0, 0, 2 \rangle$$

The curl is a constant vector pointing straight up, confirming uniform counterclockwise rotation in the xy-plane with strength 2.

• Gauss’s Law: the Divergence Theorem applied to electric fields. Total electric flux through a closed surface equals the enclosed charge.
• Faraday’s Law: Stokes’ Theorem applied to the electric field. The induced voltage around a loop equals the rate of change of magnetic flux.
• Fluid dynamics: divergence tells you where fluid is being created or destroyed. Curl tells you where it’s spinning.
• Game engines: wind simulation uses vector fields with curl for vortices and divergence for sources/sinks. Particle emitters use divergence to control outflow. Lighting on curved surfaces uses surface integrals. Flow fields guide AI pathfinding.