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
\( (\vec u \cdot \nabla) \vec v = (\vec u \cdot \nabla) v_1 \vec i + (\vec u \cdot \nabla) v_2 \vec j + (\vec u \cdot \nabla) v_3 \vec k \)
\( \quad = ( u_1 \dfrac{\partial v_1}{\partial x} + u_2 \dfrac{\partial v_1}{\partial y} + u_3 \dfrac{\partial v_1}{\partial z}\ ) \vec i \)
\( \quad\quad + ( u_1 \dfrac{\partial v_2}{\partial x} + u_2 \dfrac{\partial v_2}{\partial y} + u_3 \dfrac{\partial v_2}{\partial z}\ ) \vec j \)
\( \quad\quad\quad + ( u_1 \dfrac{\partial v_3}{\partial x} + u_2 \dfrac{\partial v_3}{\partial y} + u_3 \dfrac{\partial v_3}{\partial z}\ ) \vec k \)