8. Vector Calculus

Divergence

Divergence is a differential operator measuring the magnitude of a vector field's source or sink at a given point, defined algebraically as the dot product of the del operator with the vector field.

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

📖 Topic progress

○ Mark as complete

Definition and statement

Definition and Physical Meaning of Divergence

Let \mathbf{F}(x, y, z) = F_1 \mathbf{i} + F_2 \mathbf{j} + F_3 \mathbf{k} be a continuously differentiable vector field defined on an open set in \mathbb{R}^3, where F_1, F_2, F_3 have continuous first-order partial derivatives. The divergence of \mathbf{F}, denoted by \text{div}\,\mathbf{F} or \nabla \cdot \mathbf{F}, is defined as: $$\nabla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}$$

📐 Geometric Interpretation

Divergence measures the net rate of flux per unit volume outward from an infinitesimal control volume. A positive divergence ($\nabla \cdot \mathbf{F} > 0$) signifies a 'source' (net expansion/outflow), a negative divergence ($\nabla \cdot \mathbf{F} < 0$) signifies a 'sink' (net contraction/inflow), and a zero divergence ($\nabla \cdot \mathbf{F} = 0$) characterizes a divergence-free or solenoidal field.

⚙️ Algebraic & Structural Meaning

Algebraically, divergence acts as a contraction mapping on a vector field, reducing a tensor rank by one (transforming a vector field into a scalar field). It forms the foundational scalar kernel for the Divergence Theorem (Gauss's Theorem), bridging volume integrals to surface flux integrals.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Continuous Differentiability	`Open domain in \mathbb{R}^3`	Strictly Mandatory / Necessary	If first-order partial derivatives do not exist or are discontinuous, the partial derivatives \frac{\partial F_i}{\partial x_j} fail to exist, making \nabla \cdot \mathbf{F} undefined.
Domain Openness	`Open set U \subset \mathbb{R}^3`	Strictly Mandatory	Without an open neighborhood around each point, taking limits to define infinitesimal flux per unit volume becomes mathematically invalid.
Solenoidal Vector Field condition	`Global domain \Omega`	Essential for specialized classification	If $\nabla \cdot \mathbf{F} \neq 0$, the field is not solenoidal, and thus cannot be expressed locally as the curl of another vector potential (i.e., \mathbf{F} = \nabla \times \mathbf{A} fails).

Useful results and shortcuts

Use these identities and results when their conditions apply:

Radial Vector Field Divergence

$$ \nabla \cdot (r^n \mathbf{r}) = (n + 3) r^n $$

Rule: For position vector \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} and magnitude r = |\mathbf{r}|, the divergence of \mathbf{r}^n \mathbf{r} is instantly given by (n+3)r^n in 3D (and (n+2)r^n in 2D).

Constant Vector Dot Field

$$ \nabla \cdot (\mathbf{c} \times \mathbf{r}) = 0 $$

Rule: The divergence of the cross product of a constant vector \mathbf{c} and the position vector \mathbf{r} is identically zero because such vector fields represent rigid rotations (solenoidal fields).

Application: Direct property-based MCQs in DSSSB PGT Mathematics.

Gradient of Scalar Function Divergence

$$ \nabla \cdot (\nabla f) = \nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} $$

Rule: The divergence of a gradient field is the Laplacian of the scalar function. If f is harmonic, \nabla^2 f = 0, so the divergence is zero.

Application: Standard recurring formula in UP TGT/PGT vector calculus sections.

Linear Vector Field Shortcut

$$ \mathbf{F} = (ax + by + cz)\mathbf{i} + (dx + ey + fz)\mathbf{j} + (gx + hy + iz)\mathbf{k} \implies \nabla \cdot \mathbf{F} = a + e + i $$

Rule: For any linear vector field, the divergence is simply the trace of its Jacobian matrix (sum of the coefficients of x in x-component, y in y-component, and z in z-component).

Curl Divergence Identity

$$ \nabla \cdot (\nabla \times \mathbf{A}) = 0 $$

Rule: The divergence of the curl of any smooth vector field \mathbf{A} is always identically zero. Any curl field is automatically solenoidal.

Rapid Elimination via Solenoidal Test

$$ \sum \frac{\partial F_i}{\partial x_i} = 0 $$

Rule: When asked to identify a solenoidal field among 4 options, quickly compute the sum of partial derivatives \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}; eliminate any option where this sum is non-zero.

Common errors and counterexamples

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

⚠️ COMMON ERROR #01

Confusing Divergence with Curl

Common Pitfall: Candidates apply cross-product differential operators (curl) when divergence is requested, or vice versa, resulting in vector outputs instead of scalars.

Counterexample: \nabla \times \mathbf{F} \neq \nabla \cdot \mathbf{F}

Why it fails: Divergence is a scalar operator yielding \frac{\partial F_1}{\partial x} + \dots, whereas curl is a vector operator yielding a determinant involving \mathbf{i}, \mathbf{j}, \mathbf{k}.

🛡️ Exam Defense: Always check the dot symbol (\nabla \cdot \mathbf{F} \implies scalar) versus cross symbol (\nabla \times \mathbf{F} \implies vector).

⚠️ COMMON ERROR #02

Ignoring Singularities at the Origin

Common Pitfall: Computing \nabla \cdot \mathbf{F} = 0 everywhere for \mathbf{F} = \frac{\mathbf{r}}{r^3} without checking the origin r = 0.

Counterexample: \mathbf{F} = \frac{\mathbf{r}}{r^3} \text{ at } (0,0,0)

Why it fails: While \nabla \cdot \mathbf{F} = 0 for r > 0, the field is undefined at the origin, and application of Gauss's Theorem requires accounting for the point source at the origin.

🛡️ Exam Defense: Examine domain boundaries and origin exclusions when inverse-square or inverse-cube fields appear.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
Divergence	C1 vector field in open U \subset \mathbb{R}^3	\nabla \cdot \mathbf{F} = \sum \frac{\partial F_i}{\partial x_i}	Net outward flux per unit volume (source/sink)	Keywords: 'divergence', 'solenoidal', '\nabla \cdot'
Curl	C1 vector field in open U \subset \mathbb{R}^3	\nabla \times \mathbf{F} = \det \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \partial_x & \partial_y & \partial_z \\ F_1 & F_2 & F_3 \end{vmatrix}	Maximum circulation / axis of rotation per unit area	Keywords: 'curl', 'rotational', 'irrotational', '\nabla \times'
Gradient	C1 scalar field f on open U \subset \mathbb{R}^3	\nabla f = \frac{\partial f}{\partial x}\mathbf{i} + \frac{\partial f}{\partial y}\mathbf{j} + \frac{\partial f}{\partial z}\mathbf{k}	Direction of steepest ascent and normal to level surfaces	Keywords: 'gradient', 'scalar field', 'normal vector'
Laplacian	C2 scalar field f	\nabla^2 f = \nabla \cdot (\nabla f)	Difference between value at point and average over surrounding sphere	Keywords: 'Laplacian', 'harmonic function', '\nabla^2'

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

Find the divergence of the vector field \mathbf{F} = x^2 y \mathbf{i} + y^2 z \mathbf{j} + z^2 x \mathbf{k} at the point (1, -1, 2).

(A) 2(B) -2(C) 4(D) 0

📚 Method A: Formal Textbook Derivation

Step 1: Compute partial derivatives: \frac{\partial}{\partial x}(x^2 y) = 2xy, \frac{\partial}{\partial y}(y^2 z) = 2yz, \frac{\partial}{\partial z}(z^2 x) = 2zx. Step 2: Sum them: \nabla \cdot \mathbf{F} = 2xy + 2yz + 2zx. Step 3: Substitute (1, -1, 2): 2(1)(-1) + 2(-1)(2) + 2(2)(1) = -2 - 4 + 4 = -2... Wait, let's re-add: -2 - 4 + 4 = -2. Let's check options. If correct is -2, let's adjust: \nabla \cdot \mathbf{F} = 2xy + 2yz + 2zx = 2(-1) + 2(-2) + 2(2) = -2 - 4 + 4 = -2. Let's make option (B) -2.

⚡ Method B: 25-Second Exam Shortcut

Use Sum of partial derivatives evaluated directly at the point.

💡 Tip: Be extremely careful with signs during substitution of negative coordinates.

CANONICAL PROBLEM 02 Dual-Method Breakdown

For what value of constant k is the vector field \mathbf{F} = (x + 3y)\mathbf{i} + (y - 2z)\mathbf{j} + (x + kz)\mathbf{k} solenoidal?

(A) -2(B) -1(C) 1(D) 2

📚 Method A: Formal Textbook Derivation

Step 1: A vector field is solenoidal if \nabla \cdot \mathbf{F} = 0. Step 2: Calculate \nabla \cdot \mathbf{F} = \frac{\partial}{\partial x}(x + 3y) + \frac{\partial}{\partial y}(y - 2z) + \frac{\partial}{\partial z}(x + kz) = 1 + 1 + k = 2 + k. Step 3: Set 2 + k = 0 \implies k = -2.

⚡ Method B: 25-Second Exam Shortcut

Sum the coefficients of x in F_1, y in F_2, z in F_3: 1 + 1 + k = 0 \implies k = -2.

💡 Tip: Solenoidal condition \nabla \cdot \mathbf{F} = 0 is a staple question type in UP PGT exams.

Practice questions

Choose an answer to check it and see the solution:

MCQ 1: If \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} and r = |\mathbf{r}|, then \nabla \cdot (r \mathbf{r}) is equal to:

A 3r

B 4r

C r

D 3

Derivation & Analysis:

Using Shortcut 1, \nabla \cdot (r^n \mathbf{r}) = (n+3)r^n. Here n = 1, so (1+3)r^1 = 4r.

MCQ 2: The divergence of the vector field \mathbf{F} = yz \mathbf{i} + xz \mathbf{j} + xy \mathbf{k} is:

A xyz

B 0

C x + y + z

D 2(x + y + z)

Derivation & Analysis:

\nabla \cdot \mathbf{F} = \frac{\partial}{\partial x}(yz) + \frac{\partial}{\partial y}(xz) + \frac{\partial}{\partial z}(xy) = 0 + 0 + 0 = 0.

MCQ 3: A vector field \mathbf{F} is said to be solenoidal if:

A $\nabla \times \mathbf{F} = \mathbf{0}$

B $\nabla \cdot \mathbf{F} = 0$

C $\nabla \cdot \mathbf{F} \neq 0$

D $\nabla \times \mathbf{F} \neq \mathbf{0}$

Derivation & Analysis:

By definition, a vector field with zero divergence (\nabla \cdot \mathbf{F} = 0) is called solenoidal.

MCQ 4: If \mathbf{F} = x^3 \mathbf{i} + y^3 \mathbf{j} + z^3 \mathbf{k}, then \nabla \cdot \mathbf{F} at (1, 1, 1) is:

A 3

B 6

C 9

D 1

Derivation & Analysis:

\nabla \cdot \mathbf{F} = 3x^2 + 3y^2 + 3z^2 = 3(1) + 3(1) + 3(1) = 9.

MCQ 5: The value of \nabla \cdot \left(\frac{\mathbf{r}}{r^3}\space\right) for r \neq 0 is:

A 1

B $r^2$

C 0

D $3/r^3$

Derivation & Analysis:

Using \nabla \cdot (r^n \mathbf{r}) with n = -3: (-3 + 3)r^{-3} = 0 \cdot r^{-3} = 0.

MCQ 6: If \phi is a scalar function with continuous second partial derivatives, then \nabla \cdot (\nabla \phi) equals:

A 0

B $\nabla^2 \phi$

C $\nabla \times (\nabla \phi)$

D $\phi \mathbf{r}$

Derivation & Analysis:

\nabla \cdot (\nabla \phi) is the definition of the Laplacian of \phi, denoted by \nabla^2 \phi.

MCQ 7: If \mathbf{F} = e^x \sin y \mathbf{i} + e^x \cos y \mathbf{j} + z \mathbf{k}, then \nabla \cdot \mathbf{F} is:

A z

B 0

C 1

D $2e^x \cos y$

Derivation & Analysis:

\nabla \cdot \mathbf{F} = \frac{\partial}{\partial x}(e^x \sin y) + \frac{\partial}{\partial y}(e^x \cos y) + \frac{\partial}{\partial z}(z) = e^x \sin y - e^x \sin y + 1 = 1.

MCQ 8: Which of the following vector fields is solenoidal?

A $\mathbf{F} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$

B $\mathbf{F} = x^2\mathbf{i} + y^2\mathbf{j} + z^2\mathbf{k}$

C $\mathbf{F} = y\mathbf{i} - x\mathbf{j} + z\mathbf{k}$

D $\mathbf{F} = x\mathbf{i} - y\mathbf{j}$

Derivation & Analysis:

For \mathbf{F} = x\mathbf{i} - y\mathbf{j} in 2D, \nabla \cdot \mathbf{F} = \frac{\partial}{\partial x}(x) + \frac{\partial}{\partial y}(-y) = 1 - 1 = 0.

MCQ 9: If \mathbf{a} is a constant vector and \mathbf{r} is the position vector, then \nabla \cdot ((\mathbf{a} \cdot \mathbf{r})\mathbf{r}) is equal to:

A $4(\mathbf{a} \cdot \mathbf{r})$

B $3(\mathbf{a} \cdot \mathbf{r})$

C $\mathbf{a} \cdot \mathbf{r}$

D 0

Derivation & Analysis:

Expanding component-wise or using vector identities yields 4(\mathbf{a} \cdot \mathbf{r}).

Q10

MCQ 10: The divergence operator \nabla \cdot is:

A A vector-to-vector linear mapping

B A vector-to-scalar differential operator

C A scalar-to-vector differential operator

D A scalar-to-scalar algebraic operator

Derivation & Analysis:

Divergence takes a vector field as input and produces a scalar field as output via partial differentiation.

Common questions

No. Divergence requires a vector field as its argument. To apply divergence-like operations to a scalar field \phi, we must first take its gradient \nabla \phi (which produces a vector field) and then compute its divergence \nabla \cdot (\nabla \phi) = \nabla^2 \phi.

Divergence (\nabla \cdot \mathbf{F}) measures the net outward flux or expansion of fluid/field per unit volume from a point (scalar quantity). Curl (\nabla \times \mathbf{F}) measures the rotational tendency or vorticity of the field around a point (vector quantity).

Memorize the standard formula \nabla \cdot (r^n \mathbf{r}) = (n + 3)r^n in 3D. Instead of expanding into Cartesian coordinates \frac{x}{r}, \frac{y}{r}, etc., directly apply the exponent rule to save time and eliminate algebraic mistakes.

← Previous Topic

Directional derivatives

Next Topic →

Gauss-divergence theorem