3.2 Vector Calculus

Detailed Theory: Vector Calculus

1. Introduction to Vector Calculus

1.1 What is Vector Calculus?

Vector calculus extends calculus to vector fields - functions that assign vectors to points in space.

Key concepts:

• Scalar fields: Functions that assign scalars to points

• Vector fields: Functions that assign vectors to points

• Operations: Gradient, divergence, curl, line integrals, surface integrals

1.2 Scalar Fields

A scalar field is a function $f: \mathbb{R}^n \to \mathbb{R}$ that assigns a scalar value to each point.

Examples:

• Temperature distribution: $T(x,y,z)$

• Pressure field: $P(x,y,z)$

• Electric potential: $\phi(x,y,z)$

Notation: For 3D: $f(x,y,z)$

1.3 Vector Fields

A vector field is a function $\vec{F}: \mathbb{R}^n \to \mathbb{R}^n$ that assigns a vector to each point.

Examples:

• Velocity field of fluid flow

• Electric field $\vec{E}(x,y,z)$

• Magnetic field $\vec{B}(x,y,z)$

• Gravitational field $\vec{g}(x,y,z)$

Notation: For 3D: $\vec{F}(x,y,z) = P(x,y,z)\hat{i} + Q(x,y,z)\hat{j} + R(x,y,z)\hat{k}$

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2. Differential Operators in Vector Calculus

2.1 The Del Operator (Nabla)

a) Definition

The del operator $\nabla$ is a vector differential operator:

$$\nabla = \hat{i}\frac{\partial}{\partial x} + \hat{j}\frac{\partial}{\partial y} + \hat{k}\frac{\partial}{\partial z}$$

2.2 Gradient of a Scalar Field

a) Definition

The gradient of a scalar field $f(x,y,z)$ is:

$$\nabla f = \frac{\partial f}{\partial x}\hat{i} + \frac{\partial f}{\partial y}\hat{j} + \frac{\partial f}{\partial z}\hat{k}$$

b) Example

Find gradient of $f(x,y,z) = x^2 + y^2 + z^2$

Solution:

$$\frac{\partial f}{\partial x} = 2x, \quad \frac{\partial f}{\partial y} = 2y, \quad \frac{\partial f}{\partial z} = 2z$$

$$\nabla f = 2x\hat{i} + 2y\hat{j} + 2z\hat{k} = 2(x\hat{i} + y\hat{j} + z\hat{k})$$

2.3 Divergence of a Vector Field

a) Definition

The divergence of a vector field $\vec{F} = P\hat{i} + Q\hat{j} + R\hat{k}$ is:

$$\nabla \cdot \vec{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}$$

b) Example

Find divergence of $\vec{F} = x^2\hat{i} + y^2\hat{j} + z^2\hat{k}$

Solution:

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

2.4 Curl of a Vector Field

a) Definition

The curl of a vector field $\vec{F} = P\hat{i} + Q\hat{j} + R\hat{k}$ is:

$$\nabla \times \vec{F} = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\right)\hat{i} + \left(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}\right)\hat{j} + \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\hat{k}$$

b) Determinant Form

We can write curl as:

$$\nabla \times \vec{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{vmatrix}$$

c) Example

Find curl of $\vec{F} = y\hat{i} - x\hat{j}$

Solution:

Here $P = y$, $Q = -x$, $R = 0$

$$\nabla \times \vec{F} = \left(\frac{\partial 0}{\partial y} - \frac{\partial (-x)}{\partial z}\right)\hat{i} + \left(\frac{\partial y}{\partial z} - \frac{\partial 0}{\partial x}\right)\hat{j} + \left(\frac{\partial (-x)}{\partial x} - \frac{\partial y}{\partial y}\right)\hat{k}$$

Simplify:

$$= (0 - 0)\hat{i} + (0 - 0)\hat{j} + (-1 - 1)\hat{k} = -2\hat{k}$$

2.5 Laplacian Operator

a) Laplacian of a Scalar

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

b) Laplacian of a Vector

For $\vec{F} = P\hat{i} + Q\hat{j} + R\hat{k}$:

$$\nabla^2 \vec{F} = (\nabla^2 P)\hat{i} + (\nabla^2 Q)\hat{j} + (\nabla^2 R)\hat{k}$$

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3. Important Vector Identities

3.1 Basic Identities

1. Gradient of curl is zero: $\nabla \times (\nabla f) = \vec{0}$

2. Divergence of curl is zero: $\nabla \cdot (\nabla \times \vec{F}) = 0$

3. Divergence of gradient is Laplacian: $\nabla \cdot (\nabla f) = \nabla^2 f$

4. Curl of curl: $\nabla \times (\nabla \times \vec{F}) = \nabla(\nabla \cdot \vec{F}) - \nabla^2 \vec{F}$

3.2 Product Rules

1. Gradient of product: $\nabla(fg) = f\nabla g + g\nabla f$

2. Divergence of scalar times vector: $\nabla \cdot (f\vec{F}) = f(\nabla \cdot \vec{F}) + \vec{F} \cdot (\nabla f)$

3. Curl of scalar times vector: $\nabla \times (f\vec{F}) = f(\nabla \times \vec{F}) + (\nabla f) \times \vec{F}$

4. Divergence of cross product: $\nabla \cdot (\vec{F} \times \vec{G}) = \vec{G} \cdot (\nabla \times \vec{F}) - \vec{F} \cdot (\nabla \times \vec{G})$

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4. Line Integrals

4.1 Line Integral of a Scalar Field

a) Definition

For scalar field $f$ along curve $C$:

$$\int_C f \, ds$$

where $ds$ is arc length element.

b) Example

Evaluate $\int_C (x^2 + y^2) \, ds$ along circle $x^2 + y^2 = 4$ from $(2,0)$ to $(0,2)$

Solution:

Parameterize: $x = 2\cos t$, $y = 2\sin t$, $0 \leq t \leq \frac{\pi}{2}$

$$\frac{dx}{dt} = -2\sin t, \quad \frac{dy}{dt} = 2\cos t$$

$$ds = \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt = \sqrt{4\sin^2 t + 4\cos^2 t} dt = 2\,dt$$

$$f = x^2 + y^2 = 4\cos^2 t + 4\sin^2 t = 4$$

$$\int_C f \, ds = \int_0^{\pi/2} 4 \cdot 2\,dt = 8 \int_0^{\pi/2} dt = 8 \cdot \frac{\pi}{2} = 4\pi$$

4.2 Line Integral of a Vector Field

a) Definition

For vector field $\vec{F}$ along curve $C$:

$$\int_C \vec{F} \cdot d\vec{r}$$

b) Component Form

If $\vec{F} = P\hat{i} + Q\hat{j} + R\hat{k}$ and $d\vec{r} = dx\hat{i} + dy\hat{j} + dz\hat{k}$:

$$\int_C \vec{F} \cdot d\vec{r} = \int_C P\,dx + Q\,dy + R\,dz$$

c) Example

Evaluate $\int_C \vec{F} \cdot d\vec{r}$ where $\vec{F} = y\hat{i} - x\hat{j}$ along straight line from $(0,0)$ to $(1,1)$

Solution:

Parameterize: $x = t$, $y = t$, $0 \leq t \leq 1$

$$\vec{r}(t) = t\hat{i} + t\hat{j}, \quad \frac{d\vec{r}}{dt} = \hat{i} + \hat{j}$$

$$\vec{F}(t) = t\hat{i} - t\hat{j}$$

$$\int_C \vec{F} \cdot d\vec{r} = \int_0^1 (t\hat{i} - t\hat{j}) \cdot (\hat{i} + \hat{j}) \, dt = \int_0^1 (t - t) \, dt = \int_0^1 0 \, dt = 0$$

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5. Conservative Vector Fields

5.1 Definition

A vector field $\vec{F}$ is conservative if there exists a scalar potential function $\phi$ such that:

$$\vec{F} = \nabla \phi$$

5.2 Equivalent Conditions

For $\vec{F} = P\hat{i} + Q\hat{j} + R\hat{k}$, the following are equivalent:

1. $\vec{F}$ is conservative ($\vec{F} = \nabla \phi$)

2. $\int_C \vec{F} \cdot d\vec{r}$ is path-independent

3. $\oint_C \vec{F} \cdot d\vec{r} = 0$ for all closed curves $C$

4. $\nabla \times \vec{F} = \vec{0}$ (curl is zero)

5.3 Finding Potential Function

If $\vec{F} = \nabla \phi$, then:

$$\frac{\partial \phi}{\partial x} = P, \quad \frac{\partial \phi}{\partial y} = Q, \quad \frac{\partial \phi}{\partial z} = R$$

Integrate to find $\phi$.

5.4 Example

Check if $\vec{F} = (2xy + z^3)\hat{i} + x^2\hat{j} + 3xz^2\hat{k}$ is conservative and find potential.

Solution:

First find curl:

$$\nabla \times \vec{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ 2xy + z^3 & x^2 & 3xz^2 \end{vmatrix}$$

Compute:

$\hat{i}$ component: $\frac{\partial}{\partial y}(3xz^2) - \frac{\partial}{\partial z}(x^2) = 0 - 0 = 0$

$\hat{j}$ component: $-\left[\frac{\partial}{\partial x}(3xz^2) - \frac{\partial}{\partial z}(2xy + z^3)\right] = -[3z^2 - 3z^2] = 0$

$\hat{k}$ component: $\frac{\partial}{\partial x}(x^2) - \frac{\partial}{\partial y}(2xy + z^3) = 2x - 2x = 0$

Since $\nabla \times \vec{F} = \vec{0}$, field is conservative.

Find potential $\phi$:

From $\frac{\partial \phi}{\partial x} = 2xy + z^3$:

$$\phi = \int (2xy + z^3) dx = x^2y + xz^3 + f(y,z)$$

From $\frac{\partial \phi}{\partial y} = x^2$:

$$\frac{\partial \phi}{\partial y} = x^2 + \frac{\partial f}{\partial y} = x^2 \Rightarrow \frac{\partial f}{\partial y} = 0 \Rightarrow f = g(z)$$

So $\phi = x^2y + xz^3 + g(z)$

From $\frac{\partial \phi}{\partial z} = 3xz^2$:

$$\frac{\partial \phi}{\partial z} = 3xz^2 + g'(z) = 3xz^2 \Rightarrow g'(z) = 0 \Rightarrow g(z) = C$$

Thus $\phi = x^2y + xz^3 + C$

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6. Surface Integrals

6.1 Surface Integral of a Scalar Field

a) Definition

For scalar field $f(x,y,z)$ over surface $S$:

$$\iint_S f \, dS$$

where $dS$ is surface area element.

b) Example

Evaluate $\iint_S (x^2 + y^2) \, dS$ over sphere $x^2 + y^2 + z^2 = a^2$

Solution:

Use spherical coordinates: $x = a\sin\theta\cos\phi$, $y = a\sin\theta\sin\phi$, $z = a\cos\theta$

Surface element: $dS = a^2\sin\theta \, d\theta\,d\phi$

Also $x^2 + y^2 = a^2\sin^2\theta$

So:

$$\iint_S (x^2 + y^2) \, dS = \int_0^{2\pi}\int_0^{\pi} a^2\sin^2\theta \cdot a^2\sin\theta \, d\theta\,d\phi$$

$$= a^4 \int_0^{2\pi} d\phi \int_0^{\pi} \sin^3\theta \, d\theta$$

Using $\int_0^{\pi} \sin^3\theta \, d\theta = \frac{4}{3}$:

$$= a^4 \cdot 2\pi \cdot \frac{4}{3} = \frac{8\pi a^4}{3}$$

6.2 Surface Integral of a Vector Field (Flux)

a) Definition

For vector field $\vec{F}$ over oriented surface $S$:

$$\iint_S \vec{F} \cdot d\vec{S} = \iint_S \vec{F} \cdot \hat{n} \, dS$$

where $\hat{n}$ is unit normal vector.

b) Example

Find flux of $\vec{F} = z\hat{i} + x\hat{j} + y\hat{k}$ through surface $S$: part of plane $x + y + z = 1$ in first octant

Solution:

Surface: $z = 1 - x - y$, $x \geq 0$, $y \geq 0$, $z \geq 0$

Projection in xy-plane: triangle with vertices $(0,0)$, $(1,0)$, $(0,1)$

Normal vector: $\nabla(z - 1 + x + y) = \hat{i} + \hat{j} + \hat{k}$

Unit normal (upward): $\hat{n} = \frac{1}{\sqrt{3}}(\hat{i} + \hat{j} + \hat{k})$

Surface element: $dS = \sqrt{1 + z_x^2 + z_y^2} \, dx\,dy = \sqrt{1 + 1 + 1} \, dx\,dy = \sqrt{3} \, dx\,dy$

On surface: $\vec{F} = z\hat{i} + x\hat{j} + y\hat{k} = (1-x-y)\hat{i} + x\hat{j} + y\hat{k}$

Flux:

$$\iint_S \vec{F} \cdot \hat{n} \, dS = \iint_D \vec{F} \cdot \frac{1}{\sqrt{3}}(\hat{i} + \hat{j} + \hat{k}) \cdot \sqrt{3} \, dx\,dy$$

$$= \iint_D [(1-x-y) + x + y] \, dx\,dy = \iint_D 1 \, dx\,dy$$

Area of triangle = $\frac{1}{2} \times 1 \times 1 = \frac{1}{2}$

So flux = $\frac{1}{2}$

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7. Fundamental Theorems of Vector Calculus

7.1 Fundamental Theorem for Line Integrals

If $\vec{F} = \nabla \phi$ is conservative and $C$ is any curve from $A$ to $B$, then:

$$\int_C \vec{F} \cdot d\vec{r} = \phi(B) - \phi(A)$$

7.2 Green's Theorem (2D)

a) Statement

For simple closed curve $C$ enclosing region $D$ in xy-plane:

$$\oint_C (P\,dx + Q\,dy) = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) \, dA$$

b) Example

Use Green's theorem to evaluate $\oint_C (x^2y\,dx + y^2\,dy)$ where $C$ is triangle with vertices $(0,0)$, $(1,0)$, $(1,1)$

Solution:

Here $P = x^2y$, $Q = y^2$

$$\frac{\partial Q}{\partial x} = 0, \quad \frac{\partial P}{\partial y} = x^2$$

By Green's theorem:

$$\oint_C (x^2y\,dx + y^2\,dy) = \iint_D (0 - x^2) \, dA = -\iint_D x^2 \, dA$$

Region $D$: $0 \leq x \leq 1$, $0 \leq y \leq x$

$$-\iint_D x^2 \, dA = -\int_0^1 \int_0^x x^2 \, dy\,dx = -\int_0^1 x^2 \cdot x \, dx = -\int_0^1 x^3 \, dx = -\left[\frac{x^4}{4}\right]_0^1 = -\frac{1}{4}$$

7.3 Divergence Theorem (Gauss's Theorem)

a) Statement

For closed surface $S$ enclosing volume $V$:

$$\oiint_S \vec{F} \cdot d\vec{S} = \iiint_V (\nabla \cdot \vec{F}) \, dV$$

b) Example

Verify divergence theorem for $\vec{F} = x\hat{i} + y\hat{j} + z\hat{k}$ over sphere $x^2 + y^2 + z^2 = a^2$

Solution:

LHS (Surface integral):

On sphere, outward normal $\hat{n} = \frac{x\hat{i} + y\hat{j} + z\hat{k}}{a}$

$$\vec{F} \cdot \hat{n} = (x\hat{i} + y\hat{j} + z\hat{k}) \cdot \frac{x\hat{i} + y\hat{j} + z\hat{k}}{a} = \frac{x^2 + y^2 + z^2}{a} = \frac{a^2}{a} = a$$

Surface area = $4\pi a^2$

So flux = $a \times 4\pi a^2 = 4\pi a^3$

RHS (Volume integral):

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

Volume of sphere = $\frac{4}{3}\pi a^3$

So $\iiint_V (\nabla \cdot \vec{F}) \, dV = 3 \times \frac{4}{3}\pi a^3 = 4\pi a^3$

Both sides equal, theorem verified.

7.4 Stokes' Theorem

a) Statement

For surface $S$ with boundary curve $C$:

$$\oint_C \vec{F} \cdot d\vec{r} = \iint_S (\nabla \times \vec{F}) \cdot d\vec{S}$$

b) Example

Verify Stokes' theorem for $\vec{F} = y\hat{i} + z\hat{j} + x\hat{k}$ over hemisphere $x^2 + y^2 + z^2 = 1$, $z \geq 0$

Solution:

LHS (Line integral): Boundary is circle $x^2 + y^2 = 1$, $z = 0$

Parameterize: $x = \cos t$, $y = \sin t$, $z = 0$, $0 \leq t \leq 2\pi$

$$\vec{r}(t) = \cos t\hat{i} + \sin t\hat{j}, \quad \frac{d\vec{r}}{dt} = -\sin t\hat{i} + \cos t\hat{j}$$

On curve: $\vec{F} = y\hat{i} + z\hat{j} + x\hat{k} = \sin t\hat{i} + 0\hat{j} + \cos t\hat{k}$

$$\int_C \vec{F} \cdot d\vec{r} = \int_0^{2\pi} (\sin t\hat{i} + \cos t\hat{k}) \cdot (-\sin t\hat{i} + \cos t\hat{j}) \, dt$$

$$= \int_0^{2\pi} (-\sin^2 t) \, dt = -\int_0^{2\pi} \frac{1 - \cos 2t}{2} \, dt = -\pi$$

RHS (Surface integral): Need $\nabla \times \vec{F}$

$$\nabla \times \vec{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ y & z & x \end{vmatrix} = -\hat{i} - \hat{j} - \hat{k}$$

For hemisphere $z = \sqrt{1 - x^2 - y^2}$:

Normal vector: $\hat{n} = \frac{x\hat{i} + y\hat{j} + z\hat{k}}{z}$

Surface element: $dS = \frac{1}{z} \, dx\,dy$

$$(\nabla \times \vec{F}) \cdot \hat{n} = (-\hat{i} - \hat{j} - \hat{k}) \cdot \frac{x\hat{i} + y\hat{j} + z\hat{k}}{z} = -\frac{x + y + z}{z}$$

By symmetry, integrals of $\frac{x}{z}$ and $\frac{y}{z}$ over hemisphere are zero

So:

$$\iint_S (\nabla \times \vec{F}) \cdot d\vec{S} = \iint_D (-1) \, dx\,dy = -\text{Area of unit circle} = -\pi$$

Both sides equal, theorem verified.

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8. Applications in Physics

8.1 Fluid Dynamics

• Continuity equation: $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho\vec{v}) = 0$

• Irrotational flow: $\nabla \times \vec{v} = 0 \Rightarrow \vec{v} = \nabla \phi$

• Incompressible flow: $\nabla \cdot \vec{v} = 0$

8.2 Electromagnetism

• Gauss's law for electricity: $\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}$

• Gauss's law for magnetism: $\nabla \cdot \vec{B} = 0$

• Faraday's law: $\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$

• Ampere-Maxwell law: $\nabla \times \vec{B} = \mu_0\vec{J} + \mu_0\epsilon_0\frac{\partial \vec{E}}{\partial t}$

8.3 Heat Transfer

• Heat equation: $\frac{\partial T}{\partial t} = \alpha \nabla^2 T$

• Fourier's law: $\vec{q} = -k\nabla T$

8.4 Gravitation

• Gravitational field: $\vec{g} = -\nabla \phi$

• Gauss's law for gravity: $\nabla \cdot \vec{g} = -4\pi G\rho$

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9. Curvilinear Coordinates

9.1 Cylindrical Coordinates

• Coordinates: $(\rho, \phi, z)$

• Relations: $x = \rho\cos\phi$, $y = \rho\sin\phi$, $z = z$

Gradient:

$$\nabla f = \frac{\partial f}{\partial \rho}\hat{e}_\rho + \frac{1}{\rho}\frac{\partial f}{\partial \phi}\hat{e}_\phi + \frac{\partial f}{\partial z}\hat{e}_z$$

Divergence:

$$\nabla \cdot \vec{F} = \frac{1}{\rho}\frac{\partial}{\partial \rho}(\rho F_\rho) + \frac{1}{\rho}\frac{\partial F_\phi}{\partial \phi} + \frac{\partial F_z}{\partial z}$$

9.2 Spherical Coordinates

• Coordinates: $(r, \theta, \phi)$

• Relations: $x = r\sin\theta\cos\phi$, $y = r\sin\theta\sin\phi$, $z = r\cos\theta$

Gradient:

$$\nabla f = \frac{\partial f}{\partial r}\hat{e}_r + \frac{1}{r}\frac{\partial f}{\partial \theta}\hat{e}_\theta + \frac{1}{r\sin\theta}\frac{\partial f}{\partial \phi}\hat{e}_\phi$$

Divergence:

$$\nabla \cdot \vec{F} = \frac{1}{r^2}\frac{\partial}{\partial r}(r^2 F_r) + \frac{1}{r\sin\theta}\frac{\partial}{\partial \theta}(\sin\theta F_\theta) + \frac{1}{r\sin\theta}\frac{\partial F_\phi}{\partial \phi}$$

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10. Solved Examples for Practice

Example 1: Conservative field check

Check if $\vec{F} = (e^x\sin y)\hat{i} + (e^x\cos y)\hat{j}$ is conservative.

Solution:

For 2D: Check if $\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$

Here $P = e^x\sin y$, $Q = e^x\cos y$

$$\frac{\partial P}{\partial y} = e^x\cos y, \quad \frac{\partial Q}{\partial x} = e^x\cos y$$

Equal, so conservative.

Example 2: Divergence and curl

For $\vec{F} = x^2\hat{i} + y^2\hat{j} + z^2\hat{k}$, find $\nabla \cdot \vec{F}$ and $\nabla \times \vec{F}$

Solution:

Divergence:

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

Curl:

$$\nabla \times \vec{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ x^2 & y^2 & z^2 \end{vmatrix}$$

All cross-derivatives are 0, so $\nabla \times \vec{F} = \vec{0}$

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11. Important Formulas Summary

11.1 Differential Operators

• Gradient: $\nabla f = \frac{\partial f}{\partial x}\hat{i} + \frac{\partial f}{\partial y}\hat{j} + \frac{\partial f}{\partial z}\hat{k}$

• Divergence: $\nabla \cdot \vec{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}$

• Curl: $\nabla \times \vec{F} = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\right)\hat{i} + \left(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}\right)\hat{j} + \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\hat{k}$

11.2 Integral Theorems

• Green: $\oint_C (P\,dx + Q\,dy) = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA$

• Divergence: $\oiint_S \vec{F} \cdot d\vec{S} = \iiint_V (\nabla \cdot \vec{F}) dV$

• Stokes: $\oint_C \vec{F} \cdot d\vec{r} = \iint_S (\nabla \times \vec{F}) \cdot d\vec{S}$

11.3 Conservative Field Conditions

• $\nabla \times \vec{F} = \vec{0}$

• $\oint_C \vec{F} \cdot d\vec{r} = 0$ for all closed $C$

• $\vec{F} = \nabla \phi$ for some $\phi$

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12. Exam Tips and Common Mistakes

12.1 Common Mistakes

1. Confusing gradient, divergence, curl:

   • Gradient: scalar → vector

   • Divergence: vector → scalar

   • Curl: vector → vector

2. Forgetting parameter ranges in line/surface integrals

3. Wrong orientation for surface in flux integrals

4. Incorrect normal vector for surfaces

5. Misapplying theorems: Check conditions first

12.2 Problem-Solving Strategy

1. Identify the type: Line/surface/volume integral? Which theorem applies?

2. Check conditions: Conservative? Closed curve/surface?

3. Choose coordinates: Cartesian/cylindrical/spherical based on symmetry

4. Compute step by step: Show all work

5. Verify: Check answer has correct units/dimensions

12.3 Quick Checks

1. Gradient: Always produces vector field

2. Divergence: Always produces scalar field

3. Curl: Always produces vector field

4. Conservative: Curl must be zero

5. Incompressible: Divergence must be zero

This comprehensive theory covers all aspects of vector calculus with detailed explanations and examples, providing complete preparation for the entrance examination.