Vector Calculus Formula Sheet

Vector Fields

Definition: A vector field assigns a vector to each point in space.

$$\vec{F}(\vec{r}) = P(x,y,z)\vec{i} + Q(x,y,z)\vec{j} + R(x,y,z)\vec{k}$$

Examples: Gravity, electric fields, fluid velocity

$$\vec{g} = -g\vec{k}, \quad \vec{E} = \frac{q}{4\pi\epsilon_0 r^2}\hat{r}$$

Gradient

Definition: Rate of change of a scalar field in all directions.

$$\nabla f = \frac{\partial f}{\partial x}\vec{i} + \frac{\partial f}{\partial y}\vec{j} + \frac{\partial f}{\partial z}\vec{k}$$

Geometric meaning: Points in direction of steepest increase. Perpendicular to level surfaces.

$$D_{\vec{u}} f = \nabla f \cdot \vec{u}$$

Note: $\vec{u}$ must be a unit vector.

Divergence

Definition: Measure of outflow from a point.

$$\nabla \cdot \vec{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}$$

Physical meaning: Source (positive) or sink (negative).

$$\text{div}(\vec{F}) = \text{Tr}(D\vec{F})$$

Curl

Definition: Measure of rotation at a point.

$$\nabla \times \vec{F} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{vmatrix}$$

Physical meaning: Circulation density, angular velocity.

Laplacian Operator

On scalar functions:

$$\nabla^2 f = \Delta f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}$$

On vector fields:

$$\nabla^2 \vec{F} = \nabla(\nabla \cdot \vec{F}) - \nabla \times (\nabla \times \vec{F})$$

Vector Identities: Products

$$\nabla(fg) = f\nabla g + g\nabla f$$

$$\nabla \cdot (f\vec{F}) = f(\nabla \cdot \vec{F}) + \vec{F} \cdot \nabla f$$

$$\nabla \times (f\vec{F}) = f(\nabla \times \vec{F}) + (\nabla f) \times \vec{F}$$

$$\nabla \cdot (\vec{F} \times \vec{G}) = \vec{G} \cdot (\nabla \times \vec{F}) - \vec{F} \cdot (\nabla \times \vec{G})$$

Vector Identities: 2nd Derivatives

$$\nabla \times (\nabla f) = \vec{0}$$

$$\nabla \cdot (\nabla \times \vec{F}) = 0$$

$$\nabla \times (\nabla \times \vec{F}) = \nabla(\nabla \cdot \vec{F}) - \nabla^2\vec{F}$$

Note: Curl of gradient is zero; divergence of curl is zero.

Line Integrals: Scalar Fields

Arc length integral:

$$\int_C f \, ds = \int_a^b f(\vec{r}(t)) \|\vec{r}'(t)\| \, dt$$

Applications: Mass, charge along curve.

$$M = \int_C \rho(x,y) \, ds$$

Line Integrals: Vector Fields

Work integral:

$$\int_C \vec{F} \cdot d\vec{r} = \int_a^b \vec{F}(\vec{r}(t)) \cdot \vec{r}'(t) \, dt$$

Circulation (closed path):

$$\oint_C \vec{F} \cdot d\vec{r}$$

Conservative Fields

If curl = 0:

$$\nabla \times \vec{F} = \vec{0} \Rightarrow \text{path independent}$$ (on a simply connected domain)

Fundamental Theorem for Line Integrals:

$$\int_C \nabla f \cdot d\vec{r} = f(B) - f(A)$$

Work around closed curve: 0

Potential Functions

Scalar potential:

$$\vec{F} = \nabla f$$

Vector potential:

$$\vec{F} = \nabla \times \vec{A}$$

Laplace equation:

$$\nabla^2 f = 0 \quad (\text{harmonic})$$

Green's Theorem: Circulation

Relates line and area integrals:

$$\oint_C \vec{F} \cdot d\vec{r} = \iint_D (\nabla \times \vec{F}) \cdot \hat{k} \, dA$$

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

C: Positively oriented closed curve, D: Interior region.

Green's Theorem: Flux

Normal flux form:

$$\oint_C \vec{F} \cdot \hat{n} \, ds = \iint_D \nabla \cdot \vec{F} \, dA$$

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

Relates outward flux to divergence.

Surface Parameterization

Parametric form:

$$\vec{r}(u,v) = x(u,v)\vec{i} + y(u,v)\vec{j} + z(u,v)\vec{k}$$

Normal vector:

$$\vec{n} = \frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v}$$

Surface element:

$$dS = \left\|\frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v}\right\| du \, dv$$

Surface Integrals: Scalars

Mass element:

$$\iint_S f \, dS = \iint_D f(\vec{r}(u,v)) \left\|\frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v}\right\| du \, dv$$

Applications: Mass, charge on surface.

Surface Integrals: Flux

Flux through surface:

$$\iint_S \vec{F} \cdot d\vec{S} = \iint_S \vec{F} \cdot \hat{n} \, dS$$

$$= \iint_D \vec{F} \cdot \left(\frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v}\right) du \, dv$$

Physical meaning: Flow rate through surface.

Stokes' Theorem

Relates line and surface integrals:

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

Circulation = flux of curl through surface.

C: Boundary of S (right-hand rule), S: Oriented surface.

$$\text{Line circulation} = \text{Curl flux}$$

Divergence Theorem

Relates surface and volume integrals:

$$\iint_S \vec{F} \cdot d\vec{S} = \iiint_V \nabla \cdot \vec{F} \, dV$$

Surface flux = volume divergence.

S: Closed surface (outward normal), V: Interior volume.

Cylindrical Coordinates

Coordinates: $(r, \theta, z)$

$$x = r\cos\theta, \quad y = r\sin\theta, \quad z = z$$

Gradient:

$$\nabla f = \frac{\partial f}{\partial r}\hat{r} + \frac{1}{r}\frac{\partial f}{\partial \theta}\hat{\theta} + \frac{\partial f}{\partial z}\hat{z}$$

Divergence:

$$\nabla \cdot \vec{F} = \frac{1}{r}\frac{\partial(rF_r)}{\partial r} + \frac{1}{r}\frac{\partial F_\theta}{\partial \theta} + \frac{\partial F_z}{\partial z}$$

Spherical Coordinates

Coordinates: $(\rho, \theta, \phi)$

$$x = \rho\sin\phi\cos\theta, \quad y = \rho\sin\phi\sin\theta, \quad z = \rho\cos\phi$$

Gradient:

$$\nabla f = \frac{\partial f}{\partial \rho}\hat{\rho} + \frac{1}{\rho}\frac{\partial f}{\partial \phi}\hat{\phi} + \frac{1}{\rho\sin\phi}\frac{\partial f}{\partial \theta}\hat{\theta}$$

Volume element:

$$dV = \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta$$

Curl (Cylindrical)

Curl in cylindrical coordinates:

$$\nabla \times \vec{F} = \left(\frac{1}{r}\frac{\partial F_z}{\partial \theta} - \frac{\partial F_\theta}{\partial z}\right)\hat{r} + \left(\frac{\partial F_r}{\partial z} - \frac{\partial F_z}{\partial r}\right)\hat{\theta}$$

$$+ \frac{1}{r}\left(\frac{\partial(rF_\theta)}{\partial r} - \frac{\partial F_r}{\partial \theta}\right)\hat{z}$$

Laplacian: Other Coordinates

Cylindrical:

$$\nabla^2 f = \frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial f}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 f}{\partial \theta^2} + \frac{\partial^2 f}{\partial z^2}$$

Spherical:

$$\nabla^2 f = \frac{1}{\rho^2}\frac{\partial}{\partial \rho}\left(\rho^2\frac{\partial f}{\partial \rho}\right) + \frac{1}{\rho^2\sin\phi}\frac{\partial}{\partial \phi}\left(\sin\phi\frac{\partial f}{\partial \phi}\right) + \frac{1}{\rho^2\sin^2\phi}\frac{\partial^2 f}{\partial \theta^2}$$

Fluid Flow Applications

Continuity equation:

$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{v}) = 0$$

Incompressible flow:

$$\nabla \cdot \vec{v} = 0$$

Euler's equation:

$$\rho\left(\frac{\partial \vec{v}}{\partial t} + (\vec{v} \cdot \nabla)\vec{v}\right) = -\nabla p + \vec{f}$$

Maxwell's Equations

Gauss's law:

$$\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}$$

No magnetic monopoles:

$$\nabla \cdot \vec{B} = 0$$

Faraday's law:

$$\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$$

Ampère-Maxwell law:

$$\nabla \times \vec{B} = \mu_0\vec{J} + \mu_0\epsilon_0\frac{\partial \vec{E}}{\partial t}$$

Fundamental Theorems

Gradient Theorem (FTC for line integrals):