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

***

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

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

***

### **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: $$\nabla \cdot (\vec{F} \times \vec{G}) = \vec{G} \cdot (\nabla \times \vec{F}) - \vec{F} \cdot (\nabla \times \vec{G})$$

***

### **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: $$x = t$$, $$y = t$$, $$0 \leq t \leq 1$$

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

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

$$
\int\_C \vec{F} \cdot d\vec{r} = \int\_0^1 (t\hat{i} - t\hat{j}) \cdot (\hat{i} + \hat{j}) , dt = \int\_0^1 (t - t) , dt = \int\_0^1 0 , dt = 0
$$

***

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

***

### **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 = $$\frac{1}{2}$$

***

### **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.

***

### **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:** $$\nabla \cdot \vec{g} = -4\pi G\rho$$

***

### **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:**

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

***

### **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 $$\nabla \times \vec{F} = \vec{0}$$

***

### **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 $$\phi$$

***

### **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.