8. Vector Calculus

Vector identities

Vector identities form the algebraic bedrock of vector calculus, relating gradient, divergence, and curl through powerful differential operator expansions. This topic covers these identities.

For: UP PGT Mathematics UP TGT Mathematics DSSSB PGT / TGT KVS / NVS PGT LT Grade

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Definition and statement

Fundamental Vector Differential Operator Identities

Let $\vec{u}$ and $\vec{v}$ be differentiable vector fields, and let $\phi$ and $\psi$ be differentiable scalar fields in $\mathbb{R}^3$. Then the fundamental differential identities involving $\nabla$ are given by: (1) $\nabla (\phi \psi) = \phi \nabla \psi + \psi \nabla \phi$, (2) $\nabla \cdot (\phi \vec{v}) = \phi (\nabla \cdot \vec{v}) + (\nabla \phi) \cdot \vec{v}$, (3) $\nabla \times (\phi \vec{v}) = \phi (\nabla \times \vec{v}) + (\nabla \phi) \times \vec{v}$, (4) $\nabla \cdot (\vec{u} \times \vec{v}) = \vec{v} \cdot (\nabla \times \vec{u}) - \vec{u} \cdot (\nabla \times \vec{v})$, and (5) $\nabla \times (\vec{u} \times \vec{v}) = (\vec{v} \cdot \nabla)\vec{u} - \vec{v}(\nabla \cdot \vec{u}) - (\vec{u} \cdot \nabla)\vec{v} + \vec{u}(\nabla \cdot \vec{v})$.

📐 Geometric Interpretation

Vector identities describe how directional derivatives, fluxes, and circulations interact under scalar scaling and cross/dot products. For instance, the curl of a scalar multiple of a vector field decomposes into a circulation scaled by the scalar plus a twist induced by the spatial variation (gradient) of the scalar.

⚙️ Algebraic & Structural Meaning

They act as distributive and associative-like laws for differential operators, allowing complex vector differential expressions to be simplified, transformed, or evaluated without reverting to tedious Cartesian component expansions.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Continuity of First Partial Derivatives ($C^1$ smoothness)	`Open domain $\Omega \subset \mathbb{R}^3$`	Strictly Mandatory / Necessary	If partial derivatives are discontinuous, mixed partials may fail to commute (Clairaut's theorem fails), invalidating standard curl and divergence identities.
Differentiability of Scalar and Vector Fields	`Open domain $\Omega \subset \mathbb{R}^3$`	Strictly Mandatory	Application of differential operators $\nabla$, $\nabla \cdot$, and $\nabla \times$ becomes undefined at points where fields possess corner singularities or non-differentiable boundaries.
Twice Differentiability ($C^2$) for Second-Order Identities	`Open domain $\Omega \subset \mathbb{R}^3$`	Essential for second-order identities like $\nabla \times (\nabla \phi) = \mathbf{0}$ and $\nabla \cdot (\nabla \times \vec{v}) = 0$	Without $C^2$ continuity, mixed partial derivatives fail to be equal, meaning $\frac{\partial^2 \phi}{\partial x \partial y} \neq \frac{\partial^2 \phi}{\partial y \partial x}$, causing second-order identities to collapse.

Useful results and shortcuts

Use these identities and results when their conditions apply:

Curl of Gradient Vanishing Identity

$$ \nabla \times (\nabla \phi) = \mathbf{0} $$

Rule: The curl of any gradient field is identically zero. Instantly recognize that conservative vector fields have zero curl.

Application: Frequently asked in UP PGT to test if a given force field is conservative or irrotational.

Divergence of Curl Vanishing Identity

$$ \nabla \cdot (\nabla \times \vec{v}) = 0 $$

Rule: The divergence of any curl field (solenoidal field) is identically zero. Any vector field expressed as a curl has zero divergence.

Application: Directly tested in TGT/PGT matching questions regarding solenoidal and irrotational properties.

Position Vector $\vec{r}$ Differential Rules

$$ \nabla \cdot \vec{r} = 3, \quad \nabla \times \vec{r} = \mathbf{0}, \quad \nabla r = \frac{\vec{r}}{r} $$

Rule: Memorize standard position vector derivatives: $\nabla r^n = n r^{n-2} \vec{r}$, $\nabla \cdot (r^n \vec{r}) = (n+3)r^n$.

Laplacian of Power of $r$

$$ \nabla^2 (r^n) = n(n+1)r^{n-2} $$

Rule: Direct formula for Laplacian of $r^n$. For harmonic functions in 3D, $n = -1$, yielding $\nabla^2 (r^{-1}) = 0$ for $r \neq 0$.

Triple Vector Product Expansion (BAC-CAB Rule for Operators)

$$ \nabla \times (\nabla \times \vec{v}) = \nabla (\nabla \cdot \vec{v}) - \nabla^2 \vec{v} $$

Rule: Relates curl of curl to gradient of divergence and vector Laplacian. Essential for electromagnetic wave equations and fluid dynamics.

Application: Standard theoretical question in PGT vector calculus sections.

Rapid Dimension Substitution for Radial Fields

$$ \nabla \cdot (f(r)\vec{r}) = r f'(r) + 3f(r) $$

Rule: Replace $\nabla \cdot (f(r)\vec{r})$ instantly using the formula where 3 corresponds to spatial dimension in $\mathbb{R}^3$.

Common errors and counterexamples

These examples show how a conclusion can fail when a condition is missing:

⚠️ COMMON ERROR #01

Misapplying Vector Product Order

Common Pitfall: Writing $\nabla \times (\phi \vec{v}) = (\nabla \phi) \times (\nabla \times \vec{v})$ or forgetting that $\nabla$ acts on both terms.

Counterexample: $(\nabla \phi) \times \vec{v} \neq \phi (\nabla \times \vec{v})$

Why it fails: The product rule for vector differential operators distributes across components; missing a term leads to wrong evaluation.

🛡️ Exam Defense: Treat $\nabla$ as both a vector and a differential operator; use index notation ($\epsilon_{ijk}$) if in doubt.

⚠️ COMMON ERROR #02

Ignoring Singularity at the Origin for Radial Powers

Common Pitfall: Assuming $\nabla^2 (r^{-1}) = 0$ everywhere without checking $r = 0$.

Counterexample: $\nabla^2 \left(\frac{1}{r}\right) = -4\pi \delta(\vec{r})$

Why it fails: At $r = 0$, $1/r$ is undefined, and its Laplacian gives a Dirac delta distribution, not zero everywhere.

🛡️ Exam Defense: Always note whether the domain includes the origin $r = 0$ when dealing with negative powers of $r$.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
Gradient Identity	Scalar field $\phi \in C^1$	$\nabla (\phi \psi) = \phi \nabla \psi + \psi \nabla \phi$	Product rule for directional rates of change	Product of scalar fields
Divergence Identity	Vector field $\vec{v} \in C^1$, scalar $\phi$	$\nabla \cdot (\phi \vec{v}) = \phi (\nabla \cdot \vec{v}) + (\nabla \phi) \cdot \vec{v}$	Flux scaling and expansion	Scalar multiplied by vector field
Curl Identity	Vector field $\vec{v} \in C^1$, scalar $\phi$	$\nabla \times (\phi \vec{v}) = \phi (\nabla \times \vec{v}) + (\nabla \phi) \times \vec{v}$	Circulation scaling and twisting	Curl of scalar-vector product
Second-Order Cross Identity	Vector field $\vec{v} \in C^2$	$\nabla \times (\nabla \phi) = \mathbf{0}, \; \nabla \cdot (\nabla \times \vec{v}) = 0$	Boundary of boundary is zero equivalent	Mixed gradient/curl operations

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

If $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ and $r = |\vec{r}|$, find the value of $\nabla \cdot (r^5 \vec{r})$.

(A) $5r^5$(B) $8r^5$(C) $3r^5$(D) $5r^3$

📚 Method A: Formal Textbook Derivation

Step 1: Use the identity $\nabla \cdot (f(r)\vec{r}) = r f'(r) + 3f(r)$. Step 2: Set $f(r) = r^5$, so $f'(r) = 5r^4$. Step 3: Substitute: $r(5r^4) + 3(r^5) = 5r^5 + 3r^5 = 8r^5$.

⚡ Method B: 25-Second Exam Shortcut

Apply direct shortcut formula $\nabla \cdot (r^n \vec{r}) = (n+3)r^n$. Here $n=5$, so $(5+3)r^5 = 8r^5$.

💡 Tip: Memorize $\nabla \cdot (r^n \vec{r}) = (n+3)r^n$ to solve.

CANONICAL PROBLEM 02 Dual-Method Breakdown

Evaluate $\nabla \times (\phi \nabla \phi)$ where $\phi(x,y,z)$ is a differentiable scalar field.

(A) $\phi \nabla \phi$(B) $\mathbf{0}$(C) $\nabla^2 \phi$(D) $(\nabla \phi) \times \vec{r}$

📚 Method A: Formal Textbook Derivation

Step 1: Expand using curl product rule: $\nabla \times (\phi \nabla \phi) = \phi \nabla \times (\nabla \phi) + (\nabla \phi) \times (\nabla \phi)$. Step 2: Since $\nabla \times (\nabla \phi) = \mathbf{0}$ and cross product of a vector with itself $(\nabla \phi) \times (\nabla \phi) = \mathbf{0}$, the result is $\mathbf{0}$.

⚡ Method B: 25-Second Exam Shortcut

Curl of any gradient is zero, and cross product of identical vectors is zero.

💡 Tip: Watch for self-cross products in vector operator expressions.

Practice questions

Choose an answer to check it and see the solution:

MCQ 1: The value of $\nabla \cdot (\nabla \times \vec{A})$ for any vector field $\vec{A}$ possessing continuous second partial derivatives is:

A $1$

B $-1$

C $0$

D $\nabla^2 \vec{A}$

Derivation & Analysis:

By fundamental vector identity, the divergence of the curl of any sufficiently smooth vector field is identically zero.

MCQ 2: If $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$, then $\nabla \left(\frac{1}{r}\right)$ is equal to:

A $-\frac{\vec{r}}{r^2}$

B $-\frac{\vec{r}}{r^3}$

C $\frac{\vec{r}}{r^3}$

D $\frac{3\vec{r}}{r}$

Derivation & Analysis:

Using $\nabla(r^n) = n r^{n-2} \vec{r}$, for $n = -1$, we get $-1 r^{-3} \vec{r} = -\frac{\vec{r}}{r^3}$.

MCQ 3: The expression $\nabla \times (\nabla \phi)$ is always equal to:

A $\nabla^2 \phi$

B $\mathbf{0}$

C $1$

D $\nabla \phi$

Derivation & Analysis:

The curl of the gradient of any scalar field $\phi$ is identically zero vector $\mathbf{0}$.

MCQ 4: If $\vec{a}$ is a constant vector and $\vec{r}$ is the position vector, then $\nabla \cdot (\vec{a} \times \vec{r})$ is:

A $0$

B $\vec{a}$

C $2\vec{a}$

D $-\vec{a}$

Derivation & Analysis:

Using identity $\nabla \cdot (\vec{u} \times \vec{v}) = \vec{v} \cdot (\nabla \times \vec{u}) - \vec{u} \cdot (\nabla \times \vec{v})$, here $\vec{u} = \vec{a}$ (constant, so curl is zero) and $\vec{v} = \vec{r}$ (curl is zero). Thus $\nabla \cdot (\vec{a} \times \vec{r}) = 0$.

MCQ 5: The value of $\nabla^2 (\ln r)$ where $r = |\vec{r}|$ in 3 dimensions is:

A $\frac{1}{r^2}$

B $\frac{1}{r}$

C $0$

D $-\frac{1}{r^2}$

Derivation & Analysis:

Using $\nabla^2 (r^n) = n(n+1)r^{n-2}$ is for power functions. For $\ln r$, $\nabla (\ln r) = \frac{\vec{r}}{r^2}$, and its divergence gives $\nabla \cdot \left(\frac{\vec{r}}{r^2}\right) = \frac{1}{r^2}$.

MCQ 6: For any vector field $\vec{v}$, the vector identity for $\nabla \times (\nabla \times \vec{v})$ is:

A $\nabla (\nabla \cdot \vec{v}) - \nabla^2 \vec{v}$

B $\nabla^2 \vec{v} - \nabla (\nabla \cdot \vec{v})$

C $\nabla \times \vec{v} - \nabla^2 \vec{v}$

D $\mathbf{0}$

Derivation & Analysis:

This is the standard vector triple product expansion for differential operators: $\nabla \times (\nabla \times \vec{v}) = \nabla (\nabla \cdot \vec{v}) - \nabla^2 \vec{v}$.

MCQ 7: If $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$, then $\nabla \cdot \vec{r}$ equals:

A $0$

B $1$

C $2$

D $3$

Derivation & Analysis:

$\nabla \cdot \vec{r} = \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z} = 1 + 1 + 1 = 3$.

MCQ 8: If $\phi$ and $\psi$ are scalar fields, then $\nabla \cdot (\phi \nabla \psi)$ is equal to:

A $\phi \nabla^2 \psi + \nabla \phi \cdot \nabla \psi$

B $\phi \nabla \psi + \psi \nabla \phi$

C $\nabla \phi \cdot \nabla \psi$

D $\phi \nabla^2 \psi$

Derivation & Analysis:

Applying product rule for divergence: $\nabla \cdot (\phi \nabla \psi) = \phi (\nabla \cdot \nabla \psi) + (\nabla \phi) \cdot \nabla \psi = \phi \nabla^2 \psi + \nabla \phi \cdot \nabla \psi$.

MCQ 9: The curl of a constant vector field $\vec{c}$ is:

A $\vec{c}$

B $3\vec{c}$

C $\mathbf{0}$

D Undefined

Derivation & Analysis:

Since all partial derivatives of a constant vector field are zero, its curl is identically the zero vector $\mathbf{0}$.

Q10

MCQ 10: If $\vec{v}$ is a solenoidal vector field, then:

A $\nabla \cdot \vec{v} = 0$

B $\nabla \times \vec{v} = \mathbf{0}$

C $\nabla^2 \vec{v} = 0$

D $\vec{v} = \mathbf{0}$

Derivation & Analysis:

By definition, a solenoidal vector field has zero divergence, i.e., $\nabla \cdot \vec{v} = 0$.

Common questions

Do not memorize them by rote. Relate them to standard scalar calculus product rules (e.g., derivative of product, chain rule) and vector algebra cross/dot product expansions. Treating $\nabla$ as a formal vector greatly aids in recalling BAC-CAB and product rules.

An irrotational field has zero curl ($\nabla \times \vec{v} = \mathbf{0}$) and can be expressed as the gradient of a scalar potential. A solenoidal field has zero divergence ($\nabla \cdot \vec{v} = 0$) and can be expressed as the curl of a vector potential.

Use the direct shortcut formula $\nabla \cdot (r^n \vec{r}) = (n + 3)r^n$ for 3-dimensional space. This bypasses component-wise differentiation entirely and yields the correct option in seconds.