Vector Calculus

Vectors, coordinate systems, gradient, divergence, curl, and integral theorems

Basics

Vector

Notations

• $\bar{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$

• $\bar{B} = B_x \hat{i} + B_y \hat{j} + B_z \hat{k}$

Position vector for point $P(x,y,z) : \bar{OP} = \bar{r} = x \hat{i} + y \hat{j} + z \hat{k}$

• $r = \sqrt{x^2 + y^2 + z^2}$

Properties

1. Magnitude
2. Direction
3. Follows vector law of addition

Representations

• $\bar{A} =

  \bar{A}

  \hat{a}_A$

• $\hat{a}_A = \frac{\bar{A} }{

  \bar{A}

  }$ : unit vector

• $\bar{AB} = \bar{B} - \bar{A}$

Angle between two vectors

• $0 \le \theta_{AB} \le 180^{\circ}$

Dot Product

• $\bar{A} \cdot \bar{B} =

  \bar{A}

  \bar{B}

  \cos \theta_{AB} \rightarrow \text{scalar}$

• $\bar{A} \cdot \bar{B} = 0 \rightarrow \text{perpendicular}$

• $\bar{A} \cdot \bar{B} =

  \bar{A}

  \bar{B}

  \rightarrow \text{parallel}$

• $\bar{A} \cdot \bar{B} = -

  \bar{A}

  \bar{B}

  \rightarrow \text{anti-parallel}$

Example:

• $\bar{A} \cdot \bar{B} = A_xB_x + A_yB_y +A_zB_z$

Projection

Projection of $\bar{B}$ in the direction of $\bar{A} = \bar{B} \cdot \hat{a}_A =

\bar{B}

\hat{a}

\cos \theta_{AB} =

\bar{B}

\cos \theta_{AB}$

Cross Product

$\bar{A} \times \bar{B} =

\bar{A}

\bar{B}

\sin \theta_{AB} \hat{a}_n$

$\hat{a}_n \(: normal perpendicular to plane AB while taking curl from\)\bar{A}\(to towards\)\bar{B}$

Example:

• $\bar{A} \times \bar{B} = \begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \ A_x & A_y & A_z \ B_x & B_y &B_z \end{bmatrix}$

Physical Significance of curl:

• $\text{Area of }\triangle ABC = \frac{1}{2}

  \bar{AB} \times \bar{AC}

  $

• $\text{Area of }\square ABCD =

  \bar{AB} \times \bar{AD}

  = \frac{1}{2}

  \bar{AC} \times \bar{BD}

  $

Co-ordinate System

Cartesian Coordinate System

Variables: $(x, y, z)$

Properties

• $-\infty < x < \infty$
• $-\infty < y < \infty$
• $-\infty < z < \infty$

Cylindrical Coordinate System

Variables: $(r, \phi, z)$

Properties

• $0 \le r < \infty$
• $0 \le \phi < 2\pi$
• $-\infty < z < \infty$

Spherical Coordinate System

Variables : $(r, \theta, \phi)$

Properties

• $0 \le r < \infty$
• $0 \le \phi < 2\pi$
• $0 \le \theta \le \pi$

Vector Operator

Del/Nabla operator

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

Laplacian Operator

• $\nabla \cdot \nabla = \nabla^2 = \frac{\partial^2}{\partial{x^2}}+ \frac{\partial^2}{\partial{y^2}} + \frac{\partial^2}{\partial{z^2}}$ : scalar

Scalar-point function

• $f(x,y,z) : R^3 \rightarrow R$ (output scalar)

Level-surface equation : $f(x,y,z) = c$

Vector-point function

• $\bar{F}(x,y,z) : R^3 \rightarrow R^3$(output vector)

• $\bar{F} = F_x\hat{i} + F_y\hat{j} + F_z\hat{k}$

Differential

Gradient

Calculated for scalar-point function $f$

!!! warning “Caution”

  Convert function $f$ into the form $f = 0$ before calculating the gradient

$\text{grad} f \rightarrow \text{vector}$

$\text{grad} f = \nabla f = \nabla f(x,y,z) = (\frac{\partial{f}}{\partial{x}}, \frac{\partial{f}}{\partial{y}}, \frac{\partial{f}}{\partial{z}}) = \frac{\partial{f}}{\partial{x}}\bf{i} + \frac{\partial{f}}{\partial{y}} \bf{j} + \frac{\partial{f}}{\partial{z}} \bf{k}$ ( direction is in the dir $f > c$ )

Properties of gradient

• $

  \nabla f

  $: Maximum rate of change in the normal direction of $f$

• $\nabla(r) = \frac{\bar{r}}{r}$
• $\nabla(f(r)) = f’(r)\frac{\bar{r}}{r}$
• $\nabla^2[f(r)] = f’‘(r) + \frac{2}{r}f’(r)$

Application of gradient

1. Directional derivative (D.D.) of $f$ at point $P$ in the direction of $\bar{M} = (\nabla f)_{P} \cdot \hat{a}_M$

2. Angle between to level surface $f_1$ and $f_2$ : $\nabla f_1 \cdot \nabla f_2 =

   \nabla f_1

   \nabla f_2

   \cos \theta$

Divergence

Calculated for vector point function $\bar{F}$

• $\nabla \cdot \bar{F} \rightarrow \text{Scalar}$

• $\text{div} \bar{F} = \nabla \cdot \bar{F} = (\frac{\partial{F_x}}{\partial{x}} + \frac{\partial{F_y}}{\partial{y}} + \frac{\partial{F_z}}{\partial{z}})$

Physical significance of divergence

1 . It is only understood at a point. i.e., $(\nabla \cdot \bar{F})_P$

If $(\nabla \cdot \bar{F})_P$ := positive, $P$ is acting as a source.

If $(\nabla \cdot \bar{F})_P$ := negative, $P$ is acting as a sink.

1. Solenoidal Vector Field

   $\nabla \bar{F} = 0$, at all points $(x,y,z)$

   Example: magnetic field, any field lines that makes closed loop

Properties of divergence

• $\nabla\cdot \bar{r} = 3$

Curl

$\nabla \times \bar{F} \rightarrow \text{Vector}$

$\nabla \times \bar{F} = \begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \ \frac{\partial}{\partial{x}} & \frac{\partial}{\partial{y}} & \frac{\partial}{\partial{z}} \ F_x & F_y & F_z \end{bmatrix}$

Physical significance of curl

1. It is only understood at a point. i.e., $(\nabla \times \bar{F})_P$

   It represents the capacity of vector field to rotate the point P

2. Irrotational Vector Field/Conservative Vector Field : $(\nabla \times \bar{F})_P = \bar{0}, \forall P$

   $(\nabla \times \bar{E}) = \bar{0}$ , where $\bar{E}$ is static electric field.

Properties of curl

• $\nabla \times \bar{r} = \bar{0}$, where $\bar{r}$ is positional vector.

Null Identity

• Scalar: $\nabla \cdot [\nabla \times \bar{A}] = 0$

• Vector: $\nabla \times [\nabla \phi] = \bar{0}$

Laplacian

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

Properties of laplacian

• Rotation invariant
• $\nabla^2(fg) = f \nabla^2 g + 2(\nabla f \cdot \nabla g) + g \nabla^2 f$
• Laplace’s equation: $\nabla^2 f = 0$
• Poisson’s equation: $\nabla^2 f = - \rho$

Integral

Line Integral

$\int_C \bar{F} \cdot \bar{d}l$

Closed line encircling an Area.

Non-conservative Vector Field

• Depends on endpoints
• Path dependent
• $\oint_C \bar{F} \cdot \bar{d}l \neq 0$

Conservative Vector Field

• Depends on endpoints
• Path independent
• $\oint_C \bar{F} \cdot \bar{d}l = 0$
• $F = \nabla f$ for some scalar field
• $\int_A^B \bar{F} \cdot \bar{d}l = f(B) - f(A)$

Surface Integral

$\int_S \bar{F} \cdot \bar{d}s$ or $\int \int_S \bar{F} \cdot \bar{d}s$

Open Surface: Circle, Rectangle etc.

Closed Surface: Closed Surface Enclosing a Volume (Hollow).

Level Surface: $f(x,y,z) =constant$

Area Vector: $\bar{S} =

\bar{S}

\hat{a}_S$

$\hat{a}_S \rightarrow$ always normal to the surface

Open Surface Integral of Vector Field

$\int \int_S \bar{F} \cdot \bar{d}s = \psi$

Measure of field lines of Vector $bar{F}$ crossing surface $S$, perpendicularly = Flux

$\psi$ is flux of $\bar{F}$ through $S$

Volume Integral

$\int \int \int_V V dv$

Stoke’s Theorem

$\oint_C \bar{A} \cdot \bar{dl} = \int\int_S (\nabla \times \bar{A}) \cdot \bar{ds}$

A is continuous and differential at every point inside C

If Vector Field is irrotational then it has to be Conservative vector field. But converse is not necessarily true.

If : $\nabla \times \bar{A} = \bar{0}$ then $\oint_C \bar{A} \cdot \bar{dl} = 0$

Inverse Stoke’s Theorem :

$\int\int_S (\nabla \times \bar{A}) \cdot \bar{ds} =\oint_C \bar{A} \cdot \bar{dl}$

Green’s Theorem

$\oint_C M(x,y) dx + N(x,y) dy = \oint M dx + N dy = \int \int_R (\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y})dxdy$ if C is in anti-clockwise direction and if C is in clock-wise direction $-\int \int_R (\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y})dxdy$

M, N are continuous and differential at every point C.

Divergence Theorem

$\oint_S \bar{A} \cdot \bar{ds} = \int \int \int_V (\nabla \cdot \bar{A}) dv$