Vector calculus formulas

A scalar; $\theta \in [0,\pi ]$ is the angle between the vectors. Commutative. For non-zero vectors, $\mathbf{A}\cdot \mathbf{B}=0$ means perpendicular. $\hat{i}\cdot \hat{i}=1$, $\hat{i}\cdot \hat{j}=0$.

The scalar component of $\mathbf{A}$ along $\mathbf{B}$ is signed (negative when $\theta >90^{\circ }$); the vector projection lies along $\pm \mathbf{B}$. The direction cosines of a non-zero $\mathbf{A}$ are $A_x/|\mathbf{A}|$, $A_y/|\mathbf{A}|$, $A_z/|\mathbf{A}|$, and their squares add to 1.

A vector perpendicular to both $\mathbf{A}$ and $\mathbf{B}$, direction by the right-hand rule. Anti-commutative: $\mathbf{B}\times \mathbf{A}=-\mathbf{A}\times \mathbf{B}$. $|\mathbf{A}\times \mathbf{B}|$ is the area of the parallelogram on $\mathbf{A}$ and $\mathbf{B}$. For non-zero vectors, $\mathbf{A}\times \mathbf{B}=0$ means parallel. $\hat{i}\times \hat{j}=\hat{k}$, $\hat{j}\times \hat{k}=\hat{i}$, $\hat{k}\times \hat{i}=\hat{j}$.

Unchanged by a cyclic shift, $\mathbf{A}\cdot (\mathbf{B}\times \mathbf{C})=\mathbf{B}\cdot (\mathbf{C}\times \mathbf{A})=\mathbf{C}\cdot (\mathbf{A}\times \mathbf{B})$; changes sign when two vectors are swapped; dot and cross may be interchanged. $|[\mathbf{A}\mathbf{B}\mathbf{C}]|$ is the volume of the parallelepiped on the three vectors, and it is zero exactly when they are coplanar (linearly dependent).

The result lies in the plane of $\mathbf{B}$ and $\mathbf{C}$. The cross product is not associative: $(\mathbf{A}\times \mathbf{B})\times \mathbf{C}=(\mathbf{A}\cdot \mathbf{C})\mathbf{B}-(\mathbf{B}\cdot \mathbf{C})\mathbf{A}$, which differs in general.

Defined for a differentiable scalar field. Linear, with $\nabla (\varphi \psi )=\varphi \nabla \psi +\psi \nabla \varphi$ and $d\varphi =\nabla \varphi \cdot d\mathbf{r}$.

Rate of change of $\varphi$ per unit distance in the direction of $\mathbf{a}$. The direction must be a unit vector, so divide by $|\mathbf{a}|$ first. 'From $P$ towards the point $Q$' means $\mathbf{a}=\overrightarrow{PQ}$. 'Along the tangent to the curve $\mathbf{r}(t)$ at $t=t_0$' means $\mathbf{a}={\mathbf{r}}^{\prime }(t_0)$; 'along the normal to the surface $\psi =c$' means $\mathbf{a}=\nabla \psi$ at the point.

Because $D_{\hat{a}}\varphi =|\nabla \varphi |\cos \theta$ ($\theta$ = angle between $\hat{a}$ and $\nabla \varphi$), the rate is largest along the gradient, most negative opposite to it, and zero along any direction tangent to the level surface.

Valid where $\nabla \varphi \ne 0$. The sign picks a side; for a closed surface (a sphere, say) the outward normal points away from the enclosed region. For $z=g(x,y)$ take $\varphi =z-g(x,y)$, so $\nabla \varphi =-g_x\hat{i}-g_y\hat{j}+\hat{k}$, which points upward.

The angle between the surfaces ${\varphi }_1=c_1$ and ${\varphi }_2=c_2$ at a common point is the angle between their normals there, both gradients evaluated at that point. Each normal is fixed only up to sign, so the angle between the surfaces is usually quoted as the acute one: take the absolute value of the numerator. They cut orthogonally when $\nabla {\varphi }_1\cdot \nabla {\varphi }_2=0$.

Valid for $r\ne 0$. Examples: $\nabla (r^2)=2\mathbf{r}$ and $\nabla (1/r)=-\mathbf{r}/r^3$.

A scalar field. $\nabla \cdot \mathbf{r}=3$; a constant field has zero divergence.

A vector field. Expand the determinant along the first row; the $\hat{j}$ term, written here in its cyclic form, is where sign slips happen. $\nabla \times \mathbf{r}=0$.

Equivalent for a continuous $\mathbf{F}$ on any connected open region. If $\mathbf{F}$ also has continuous first partial derivatives, each implies $\nabla \times \mathbf{F}=0$. The converse, curl zero implies conservative, needs $\mathbf{F}$ to have continuous first partial derivatives on a simply connected region (every closed curve in it can be shrunk to a point without leaving it).

For any piecewise smooth path from $A$ to $B$ on which $\varphi$ is continuously differentiable. Sign convention: here a potential means $\mathbf{F}=\nabla \varphi$; electromagnetics writes $\mathbf{E}=-\nabla V$, which flips the sign of the potential but not the test.

On a simply connected region with $P$, $Q$ continuously differentiable. This is the $\hat{k}$ component of the curl test for $\mathbf{F}=P\hat{i}+Q\hat{j}$.

Every gradient field is irrotational and every curl field is solenoidal. So a field with non-zero divergence cannot be the curl of anything, and a field with non-zero curl cannot be a gradient.

The order in $\nabla \varphi \times \mathbf{F}$ matters: swapping the factors flips the sign.

A scalar. $\varphi$ is harmonic when ${\nabla }^2\varphi =0$ (Laplace's equation). If $\mathbf{F}=\nabla \varphi$, then $\nabla \cdot \mathbf{F}={\nabla }^2\varphi$.

Here ${\nabla }^2\mathbf{F}={\nabla }^2{F}_1\hat{i}+{\nabla }^2{F}_2\hat{j}+{\nabla }^2{F}_3\hat{k}$, taking the Laplacian component by component; this componentwise rule holds for Cartesian components only. For a solenoidal field it reduces to $\nabla \times (\nabla \times \mathbf{F})=-{\nabla }^2\mathbf{F}$.

Valid for $r\ne 0$. Special cases: $\nabla \cdot \mathbf{r}=3$; $\nabla \cdot (\mathbf{r}/r^3)=0$ and ${\nabla }^2(1/r)=0$ away from the origin.

Depends on orientation: reversing the direction of travel changes the sign. It does not depend on which parametrisation is used, as long as the direction is the same.

Independent of orientation, because $ds>0$. With $f=1$ it gives the length of $C$; with $f$ a line density it gives the mass of a wire.

The circle parametrisation runs counter-clockwise, which keeps the enclosed disc on the left of the direction of travel. For the ellipse $x^2/a^2+y^2/b^2=1$ use $x=a\cos t$, $y=b\sin t$.

${\mathbf{r}}_u\times {\mathbf{r}}_v$ is a normal whose length is the area scale factor, so $dS=|{\mathbf{r}}_u\times {\mathbf{r}}_v|dudv$. If it points to the wrong side, use ${\mathbf{r}}_v\times {\mathbf{r}}_u$, which flips the sign. For a scalar $f$, ${\iint }_{S}fdS={\iint }_{D}f(\mathbf{r}(u,v))|{\mathbf{r}}_u\times {\mathbf{r}}_v|dudv$; with $f=1$ this is the area of $S$, and it does not depend on the side chosen.

The flux formula is for the upward normal ($\hat{n}\cdot \hat{k}>0$); negate it for the downward normal. In general, projecting onto the $xy$-plane uses $dS=\frac{dxdy}{|\hat{n}\cdot \hat{k}|}$, valid where $\hat{n}\cdot \hat{k}\ne 0$.

Outward normals. $\theta$ is measured from the $+z$ axis ($0\le \theta \le \pi$) and $\phi$ around it ($0\le \phi <2\pi$). Areas: $4\pi a^2$ for the sphere, $2\pi ah$ for the curved side of a cylinder of height $h$.

Cartesian, cylindrical and spherical. Volumes: sphere $\frac{4}{3}\pi a^3$, cylinder $\pi a^{2}h$, cone $\frac{1}{3}\pi a^{2}h$.

$C$ is a piecewise smooth simple closed curve bounding $R$, traversed counter-clockwise; $P$, $Q$ and their first partial derivatives are continuous on an open region containing $R$. Clockwise traversal changes the sign. For a region with holes, the outer boundary runs counter-clockwise and each inner boundary clockwise.

Green's theorem with $\partial Q/\partial x-\partial P/\partial y=1$, $C$ counter-clockwise. For the ellipse $x=a\cos t$, $y=b\sin t$ it gives $\pi ab$.

$S$ is a piecewise smooth oriented surface whose boundary is the closed curve $C$; $\mathbf{F}$ has continuous first partial derivatives on a region containing $S$. Right-hand rule: with the thumb along $\hat{n}$, the fingers curl in the direction of $C$ (for $\hat{n}=\hat{k}$, $C$ runs counter-clockwise seen from above). Every surface that meets these conditions and has the same boundary and orientation gives the same value.

$S$ is a closed piecewise smooth surface enclosing the volume $V$, $\hat{n}$ is the outward unit normal, and $\mathbf{F}$ has continuous first partial derivatives throughout $V$ (no singularity inside).