# Vector Calculus Formulas and Identities

 The "del" ($\nabla$) operator is defined in terms of partial derivatives as follows
$\nabla = \left( \dfrac{\partial }{\partial x} , \dfrac{\partial }{\partial y} , \dfrac{\partial }{\partial z} \right)$

## Gradient

If $\psi$ is a function of the variables $x , y$ and $z$ , the gradient is a vector defined by
$\text{grad} \psi = \nabla \psi = \left(\dfrac{\partial \psi}{\partial x},\dfrac{\partial \psi }{\partial y},\dfrac{\partial \psi}{\partial z}\right)$

## Divergence

If $\left( v_1(x,y,z) , v_2(x,y,z) , v_3(x,y,z) \right)$ are the components of vector $\vec v$, the divergence of vector $\vec v$ is a scalar defined by
$\text{div} \; \vec v = \nabla \cdot \vec v = \dfrac{\partial v_1}{\partial x} + \dfrac{\partial v_2 }{\partial y} + \dfrac{\partial v_3}{\partial z}$

## Curl

If $\left( v_1(x,y,z) , v_2(x,y,z) , v_3(x,y,z) \right)$ are the components of vector $\vec v$, the curl of vector $\vec v$ is a vector defined by
$\text{curl} \; \vec v = \nabla \times \vec v = \begin{vmatrix} \vec i & \vec j &\vec k\\ \dfrac{\partial }{\partial x} & \dfrac{\partial }{\partial y} & \dfrac{\partial }{\partial z} \\ v_1 & v_2 & v_3 \\ \end{vmatrix}$

$= \left(\dfrac{\partial v_3}{\partial y} - \dfrac{\partial v_2}{\partial z}\right) \vec i - \left(\dfrac{\partial v_3}{\partial x} - \dfrac{\partial v_1}{\partial z}\right) \vec j + \left(\dfrac{\partial v_2}{\partial x} - \dfrac{\partial v_1}{\partial y}\right) \vec k$

## Laplacian for a Scalar

If $\psi$ is a function of the variables $x , y$ and $z$ , the Laplacian of $\psi$ is a scalar defined by
$\nabla^2 \psi = \text{div} \; \text{grad} \; \psi = \nabla \cdot \nabla \psi = \dfrac{\partial^2 \psi}{\partial x^2} + \dfrac{\partial^2 \psi}{\partial y^2} + \dfrac{\partial^2 \psi}{\partial z^2}$

## Laplacian for a Vector

If $\left( v_1(x,y,z) , v_2(x,y,z) , v_3(x,y,z) \right)$ are the components of vector $\vec v$, the Laplacian of vector $\vec v$ is a vector defined by
$\nabla^2 \vec v = \text{grad} \; \text{div} \; \vec v - \text{curl} \; \text{curl} \; \vec v = \nabla (\nabla \cdot \vec v) - \nabla \times (\nabla \times \vec v)$
$\quad = \nabla^2 v_1 \vec i + \nabla^2 v_2 \vec j + \nabla^2 v_3 \vec k$
$\quad = (\dfrac{\partial^2 v_1}{\partial x^2} + \dfrac{\partial^2 v_1}{\partial y^2} + \dfrac{\partial^2 v_1}{\partial z^2}) \vec i$
$\quad \quad +(\dfrac{\partial^2 v_2}{\partial x^2} + \dfrac{\partial^2 v_2}{\partial y^2} + \dfrac{\partial^2 v_2}{\partial z^2}) \vec j$
$\quad \quad \quad \quad + (\dfrac{\partial^2 v_3}{\partial x^2} + \dfrac{\partial^2 v_3}{\partial y^2} + \dfrac{\partial^2 v_3}{\partial z^2}) \vec k$

## Identities

In what follows, $\psi$ is a function and $\vec v$ is a vector
$\text{div} \; \text{curl} \; \vec v = \nabla \cdot (\nabla \times \vec v) = 0$
$\text{curl} \; \text{grad} \; \psi = \nabla \times (\nabla \; \psi) = 0$

## Gradient, Divergence and Curl of Product

In what follows, $\psi$ and $\phi$ are functions and $\vec v$ is a vector
$\text{grad} \; (\psi \phi) = \psi \; (\nabla \phi) + \phi \; (\nabla \psi)$
$\text{div} \; (\psi \vec v) = \psi \; (\nabla \cdot \vec v) + \vec v \cdot (\nabla\psi)$
$\text{curl} \; (\psi \vec v) = \psi \; ( \nabla \times \vec v) + \nabla \psi \times \vec v$
$\text{grad} \; (\vec u \cdot \vec v) = \vec u \times (\nabla \times \vec v) + \vec v \times (\nabla \times \vec u)+ (\vec u \cdot \nabla) \vec v + (\vec v \cdot \nabla ) \vec u$
$\text{div} \; (\vec u \times \vec v) = (\nabla \times \vec u) \cdot \vec v - (\nabla \times \vec v) \cdot \vec u$
$\text{curl} \; (\vec u \times \vec v ) = \vec u (\nabla \cdot \vec v) - \vec v (\nabla \cdot \vec u) + (\vec v \cdot \nabla) \vec u - (\vec u \cdot \nabla) \vec v$
NOTE
In the above formulas, $\vec u \cdot \nabla$ may be considered as a scalar operator and is given by
$\vec u \cdot \nabla = u_1 \dfrac{\partial }{\partial x} + u_2 \dfrac{\partial }{\partial y} + u_3 \dfrac{\partial }{\partial z}\$
If it is applied to scalar function $f$, it gives a scalar
$(\vec u \cdot \nabla) f = u_1 \dfrac{\partial f}{\partial x} + u_2 \dfrac{\partial f}{\partial y} + u_3 \dfrac{\partial f}{\partial z}\$
If it is applied to vector function $\vec v$, it gives a vector
$(\vec u \cdot \nabla) \vec v = (\vec u \cdot \nabla) v_1 \vec i + (\vec u \cdot \nabla) v_2 \vec j + (\vec u \cdot \nabla) v_3 \vec k$
$\quad = ( u_1 \dfrac{\partial v_1}{\partial x} + u_2 \dfrac{\partial v_1}{\partial y} + u_3 \dfrac{\partial v_1}{\partial z}\ ) \vec i$
$\quad\quad + ( u_1 \dfrac{\partial v_2}{\partial x} + u_2 \dfrac{\partial v_2}{\partial y} + u_3 \dfrac{\partial v_2}{\partial z}\ ) \vec j$
$\quad\quad\quad + ( u_1 \dfrac{\partial v_3}{\partial x} + u_2 \dfrac{\partial v_3}{\partial y} + u_3 \dfrac{\partial v_3}{\partial z}\ ) \vec k$

## More References and Links

Handbook of Mathematical Functions Engineering Mathematics with Examples and Solutions