Vector Calculus Equations and definitions

Definition

Formula sheet for the basic vector calculus operators and the big conversion theorems.

Setup

In Cartesian coordinates, the Del operator ∇ is:

$$\nabla =\hat{\mathbf{i}}\frac{\partial }{\partial x}+\hat{\mathbf{j}}\frac{\partial }{\partial y}+\hat{\mathbf{k}}\frac{\partial }{\partial z}$$

For a scalar field $f(x,y,z)$ and vector field:

$$\mathbf{F}=(P,Q,R)$$

the common operators are below.

Gradient

Scalar field to vector field:

$$\nabla f=(f_x,f_y,f_z)$$

Meaning: direction and rate of fastest increase.

Divergence

Vector field to scalar field:

$$\nabla \cdot \mathbf{F}=P_x+Q_y+R_z$$

Meaning: local source/sink strength.

Curl

Vector field to vector field:

$$\nabla \times \mathbf{F}=(R_y-Q_z,P_z-R_x,Q_x-P_y)$$

Meaning: local rotation.

Laplacian

Scalar field to scalar field:

$${\nabla }^{2}f=\nabla \cdot (\nabla f)=f_{xx}+f_{yy}+f_{zz}$$

Meaning: local difference from nearby values.

Integrals

Scalar line integral:

$${\int }_{C}fds$$

Adds up scalar values along a curve.

Vector line integral:

$${\int }_C\mathbf{F}\cdot d\mathbf{r}$$

Adds up how much the field pushes along a curve.

Flux through a surface $S$:

$${\iint }_S\mathbf{F}\cdot \mathbf{n}dS$$

Adds up how much field passes through the surface.

Big theorems

These are the “turn hard local stuff into boundary stuff” results.

Gradient theorem

$${\int }_C\nabla f\cdot d\mathbf{r}=f(B)-f(A)$$

For conservative fields, only endpoints matter.

Divergence theorem

$${\iiint }_V\nabla \cdot \mathbf{F}dV={\iint }_{\partial V}\mathbf{F}\cdot \mathbf{n}dS$$

Total source inside a volume equals total flux out through the boundary.

Stokes’ theorem

$${\iint }_S(\nabla \times \mathbf{F})\cdot \mathbf{n}dS={\oint }_{\partial S}\mathbf{F}\cdot d\mathbf{r}$$

Total curl through a surface equals circulation around the boundary.

This is why vector calculus is all over Electromagnetism index, fluid dynamics, diffusion, and Force-free and NLFFF coronal fields.

Quick type check

• $\nabla f$: scalar to vector
• $\nabla \cdot \mathbf{F}$: vector to scalar
• $\nabla \times \mathbf{F}$: vector to vector
• ${\nabla }^{2}f$: scalar to scalar

This catches a lot of mistakes before the algebra gets messy.