Vector Calculus Identities

Vectors are fundamental tools in engineering, mathematics, and physics. They are widely used in mechanics, electromagnetism, fluid dynamics, and computer graphics. Vector identities are special algebraic relations involving vector differential operators such as gradients (∇), divergence (∇⋅), curl (∇×), and Laplacian (∇²).

Vector identities summarize important relations between gradient, divergence, curl, and Laplacian operators used to simplify vector calculus computations.

Basic Vector Identities are given below:

Gradient Identities

The gradient of a scalar function f is a vector that points in the direction of the greatest rate of increase of f and whose magnitude is the rate of increase by ∇f.

• ∇(f+g) = ∇f + ∇g
• ∇(cf) = c∇f, where c is a constant
• ∇(fg) = f∇g + g∇f
• ∇(gf​)=g2g∇f−f∇g​,g(x)
• \nabla\left(\frac{f}{g}\right) = \frac{g \nabla f - f \nabla g}{g^2}, \quad g(x) \neq 0
• \nabla (\vec{F} \cdot \vec{G}) = \vec{F} \times (\nabla \times \vec{G}) - (\nabla \times \vec{F}) \times \vec{G} + (\vec{G} \cdot \nabla) \vec{F} + (\vec{F} \cdot \nabla) \vec{G}

Divergence Identities

The divergence measures how much a vector field spreads out from a point.

• ∇.( \overrightarrow{F} + \overrightarrow{G}) = ∇. \overrightarrow{F} + ∇. \overrightarrow{G}
• ∇. (c\overrightarrow{F}) = c(∇ . \overrightarrow{F}), where c is a constant
• ∇⋅(f \overrightarrow{F})=f(∇⋅\overrightarrow{F}) + \overrightarrow{F}⋅∇f
• ∇⋅(\overrightarrow{F} × \overrightarrow{G}) = \overrightarrow{G}⋅(∇ × \overrightarrow{F}) −\overrightarrow{F}⋅(∇ × \overrightarrow{G})

Curl Identities

The curl measures the rotation or “circulation” of a vector field at a point.

• ∇.( \overrightarrow{F} + \overrightarrow{G}) = ∇. \overrightarrow{F} + ∇. \overrightarrow{G}
• ∇. (c\overrightarrow{F}) = c(∇ . \overrightarrow{F}), where c is a constant
• ∇⋅(f \overrightarrow{F})=f(∇⋅\overrightarrow{F}) + ∇f. \overrightarrow{F}
• ∇⋅(\overrightarrow{F} × \overrightarrow{G}) = \overrightarrow{F}(∇ . \overrightarrow{G}) - \overrightarrow{G}(∇. \overrightarrow{F}) + ( \overrightarrow{G}.∇) \overrightarrow{F}) - (\overrightarrow{F}⋅∇)\overrightarrow{G})

Laplacian Identities

The Laplacian measures the rate at which the average value of a function around a point differs from its value at that point:

• ∇²(f+g) = ∇²f + ∇²g
• ∇²(cf) = c∇²f,   where c is a constant.
• ∇²(fg) = f∇²g + 2(∇f⋅∇g) + g∇²f

Other Identities

• ∇. (∇ x \overrightarrow{F}) = 0
• ∇ × (∇f) = 0
• ∇⋅(∇f × ∇g) = 0
• ∇⋅(f∇g − g∇f) = f∇²g − g∇²f

Solved Questions on Vectors Identities

Question 1: Compute the gradient of the product of two scalar functions f and g, where f(x, y, z) = x² and g(x, y, z) = y².

Solution:

∇(fg) = f∇g + g∇f

∇f=(∂f/∂x, ∂f/∂y, ∂f/∂z) = (2x, 0, 0)

g = (0, 2y, 0)

∇(fg) = x²(0, 2y, 0) + y²(2x, 0, 0) = (2xy²,2x²y, 0)

Question 2: Compute the divergence of the scalar-vector product f \overrightarrow{F} , where f(x,y,z)=x and \overrightarrow{F}(x,y,z) = (y,z,x).

Solution:

Using the identity:

\nabla \cdot \overrightarrow{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} = 0

The divergence of \overrightarrow{F}is:

\nabla \cdot \vec{F} = \frac{\partial y}{\partial x} + \frac{\partial z}{\partial y} + \frac{\partial x}{\partial z} = 0∇⋅F=∂x∂y​+∂y∂z​+∂z∂x​=0

The gradient of f is:

∇f = (1, 0, 0)

Dot product:

\overrightarrow{F} .∇ f = (y, z, x)⋅(1, 0, 0) = y

Therefore:

\nabla \cdot (f\overrightarrow{F}) = x \cdot 0 + y = y∇⋅(fF) = x⋅0 + y = y

Question 3: Verify that the curl of the gradient of a scalar function is zero, i.e., show ∇×(∇f) = 0 for f(x, y, z) = xyz.

Solution:

∇f = (yz, xz, xy)

\nabla \times (\nabla f) =\begin{vmatrix}\hat{i} & \hat{j} & \hat{k} \\\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\\frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} & \frac{\partial f}{\partial z}\end{vmatrix} = \vec{0}

All components cancel, so ∇×(∇f) = 0.

Question 4: Compute the Laplacian of the product of two scalar functions, ∇²(fg), where f(x,y,z) = x and g(x, y, z) = y.

Solution:

∇²(fg) = f∇²g + 2(∇f⋅∇g) + g∇²f∇f = (1, 0, 0),∇g = (0, 1, 0)

∇²f = 0,∇²g = 0

∇²(fg) = x(0)+ 2(1⋅0+0⋅1+0⋅0) + y(0) = 0

Unsolved Questions on Vector Identities

Question 1: Find the gradient of the scalar field: f(x, y, z) = e^xysin⁡ z

Question 2: Compute the Laplacian of the scalar field: f(x, y, z) = ln⁡(x² + y² + z²).

Question 3: Compute the divergence of the vector field: \vec{A}(x,y,z) = x^2 y \, \hat{i} + y z^2 \, \hat{j} + x z \, \hat{k}

Question 4: Find ∇²(fg) if f = x² + y², g = e^z.