Vector Analysis

Abstract

Several complex problems in electromagnetism involve many physical quantities having both magnitude and direction.

Exercises

Problem 1.1

Check whether the two vectors ${\mathbf{P}} = 12\,{\hat{\mathbf{i}}} - 6\,{\hat{\mathbf{j}}} + 3\,{\hat{\mathbf{k}}}$ and ${\mathbf{Q}}\, = \,{\hat{\mathbf{i}}} - {\hat{\mathbf{j}}} - z\,{\hat{\mathbf{k}}}$ are perpendicular to each other.

Problem 1.2

If ${\mathbf{P}} = 1 0{\hat{\mathbf{i}}} - 4{\hat{\mathbf{j}}} + 2{\hat{\mathbf{k}}}$ find Q if Q is parallel to P.

Problem 1.3

Check whether the vectors ${\mathbf{P}} = 2{\hat{\mathbf{i}}} + 4{\hat{\mathbf{j}}} + 18{\hat{\mathbf{k}}}$,

${\mathbf{Q}} = 9{\hat{\mathbf{i}}} + 3{\hat{\mathbf{j}}} + 6{\hat{\mathbf{k}}}$ and ${\mathbf{R}} = 10{\hat{\mathbf{i}}} + 5{\hat{\mathbf{j}}} + 15{\hat{\mathbf{k}}}$ are coplanar.

Problem 1.4

If ${\mathbf{P}} = 3{\hat{\mathbf{i}}} + 2{\hat{\mathbf{j}}} + 5{\hat{\mathbf{k}}}$ find the unit vector in the direction of P.

Problem 1.5

Check whether the three vectors ${\mathbf{A}} = 2{\hat{{\mathbf{i}}}} + 3{\hat{{\mathbf{j}}}} + 5{\hat{{\mathbf{k}}}}$,

and ${\mathbf{B}} = {\hat{{\mathbf{i}}}} + {\hat{{\mathbf{j}}}} - {\hat{{\mathbf{k}}}}$, ${\mathbf{C}} = 3{\hat{{\mathbf{i}}}} + 4{\hat{{\mathbf{j}}}} + 4{\hat{{\mathbf{k}}}}$ form a right angled triangle.

Problem 1.6

A vector P is directed from (3, 4, 5) to (1, 3, 2). Determine $\left| {\mathbf{P}} \right|$ and unit vector in the direction of P.

Problem 1.7

Find the angle between the two vectors ${\mathbf{P}} = {\hat{{\mathbf{i}}}} - {\hat{{\mathbf{j}}}} + {\hat{{\mathbf{k}}}}$ and ${\mathbf{Q}} = 3{\hat{{\mathbf{i}}}} - 2{\hat{{\mathbf{j}}}} - {\hat{{\mathbf{k}}}}$ using dot product and cross product.

Problem 1.8

Find the value of a such that the three vectors $1 0{\hat{{\mathbf{i}}}} - 5{\hat{{\mathbf{j}}}} + 5{\hat{{\mathbf{k}}}}$,

$2{\hat{{\mathbf{i}}}} + 4{\hat{{\mathbf{j}}}} - 6{\hat{{\mathbf{k}}}}$ and $3{\hat{{\mathbf{i}}}} + 2{\hat{{\mathbf{j}}}} + a{\hat{{\mathbf{k}}}}$ are coplanar.

Problem 1.9

Find the projection of vector ${\mathbf{P}} = {\hat{{\mathbf{i}}}} + 3{\hat{{\mathbf{j}}}} + 2{\hat{{\mathbf{k}}}}$ on vector ${\mathbf{Q}} = 2{\hat{{\mathbf{i}}}} + {\hat{{\mathbf{j}}}} - 3{\hat{{\mathbf{k}}}}$

Problem 1.10

If P + Q + R = 0 then show that P × Q = Q × R = R × P.

Problem 1.11

Calculate (P × Q) ∙ (R × S)

Problem 1.12

Show that ${\mathbf{\nabla }}\; \log \;\text{r} = \frac{{\mathbf{r}}}{{\text{r}^{2} }}$ where r is the position vector.

Problem 1.13

If r is the position vector show that

• (i) ${\mathbf{\nabla }}\text{r}^{\text{n}} = \text{rn}^{{\text{n}} - 2} {\mathbf{r}}$

• (ii) div r = 3

• (iii) $\text{div}\,\text{r}^{\text{n}} {\mathbf{r}} = ( \text{n + 3)r}^{\text{n}}$

• (iv) $\text{curl}(\text{r}^{\text{n}} {\mathbf{r}}) = 0$

Problem 1.14

Calculate ${\mathbf{\nabla }}\,\text{u}\,\text{at (1,} - 5 , \text{ 3)\,if}\,\text{u} = x^ 2z + y^ 3z^ 2$.

Problem 1.15

Show that ${\mathbf{\nabla }}\,\cdot\,({\mathbf{P}} + {\mathbf{Q}}) = {\mathbf{\nabla }}\,\cdot\,{\mathbf{P}} + {\mathbf{\nabla }}\,\cdot\,{\mathbf{Q}}$

Problem 1.16

Calculate a if ${\mathbf{A}} = (3x + y){\hat{{\mathbf{i}}}} + (z - x){\hat{{\mathbf{j}}}} + (\text{a}y + z){\hat{{\mathbf{k}}}}$ is solenoidal.

Problem 1.17

If ${\mathbf{P}} = 3x^ 2{\hat{\mathbf{i}}} - y^ 2z{\hat{\mathbf{j}}} + 2x^ 3z{\hat{\mathbf{k}}}$ find

• (a) ${\mathbf{\nabla }} \times {\mathbf{P}}$

• (b) ${\mathbf{\nabla }} \times ({\mathbf{\nabla }} \times {\mathbf{P}})$

Problem 1.18

If ${\mathbf{P}} = ( 2x^ 2\text{ + }y ){\hat{\mathbf{i}}} - 1 0yz{\hat{\mathbf{j}}} + 1 2xz^{2} {\hat{\mathbf{k}}}$

Calculate $\int {{\mathbf{P}}\,\cdot\,\text{d}{\mathbf{r}}}$ along the straight line from (0, 0, 0) to (1, 1, 0) and then to (1, 1, 1).

Problem 1.19

Calculate the gradients of

• (a) $\text{u}(\text{r},\uptheta,\upphi) = 3 \text{r}\, \sin \,\uptheta - 4\upphi\text{ + 1}$

• (b) $\text{u}(\text{r},\uptheta,z) = 3 \text{cos}\,\upphi - \text{r}z$

Problem 1.20

Calculate div P if

• (a) ${\mathbf{P}} = 3{\hat{\mathbf{r}}} + \text{r}\, \sin \,\uptheta\,{\hat{\varvec{\uptheta}}} + \text{r}{\hat{\varvec{\upphi}}}$

• (b) ${\mathbf{P}} = \text{r}{\hat{\mathbf{r}}} + z \cos \,\upphi\,{\hat{\varvec{\upphi}}} + 3{\hat{\mathbf{z}}}$

Problem 1.21

Calculate the net flux of the vector field ${\mathbf{P}}(x ,y ,z) = 3xy^ 2{\hat{\mathbf{i}}} + z^ 2{\hat{\mathbf{j}}} + y^ 3{\hat{\mathbf{k}}}$ emerging from a cube of dimensions $0 \le x ,y ,z \le 1$

Problem 1.22

Solve Problem 1.21 using Gauss’s divergence theorem.

Problem 1.23

Verity divergence theorem for the vector ${\mathbf{P}} = x^{3} {\hat{\mathbf{i}}} + y^ 3{\hat{\mathbf{j}}} + z^ 3{\hat{\mathbf{k}}}$ for the cube $0 \le x ,y ,z \le 1$ shown in Fig. 1.25.

Problem 1.24

Verify Stoke’s theorem for the vector ${\mathbf{P}} = (x\text{ + }y ){\hat{\mathbf{i}}} + (y\text{ + }z ){\hat{\mathbf{j}}} + (x\text{ + }z ){\hat{\mathbf{k}}}$ for a plane rectangular area with vertices (0, 0), (2, 0), (2, 1), (0, 1) as shown in Fig. 1.46.

Fig. 1.46

A rectangular area in Cartesian coordinates (Problem 1.24)

Problem 1.25

Find the value of a, b and c if

$$ {\mathbf{P}} = {(4}x{ + 5}y{ + \text{a}}\,z{)}{\hat{\mathbf{i}}} + {(3}x{ + \text{b y} + }z {)}{\hat{\mathbf{j}}} + ( \text{c}\,x{ + 2}y - {3}z{)}\,{\hat{\mathbf{k}}}\,{\text{is}}\,{\text{irrotational}} . $$

Problem 1.26

Calculate the surface area and volume of a sphere by integrating surface and volume elements in spherical polar coordinates.

Problem 1.27

Calculate the surface area and volume of a cylinder by integrating surface and volume elements in cylindrical coordinates.

Problem 1.28

Prove that for any vector ${\mathbf{G}}\int\limits_{\uptau} {{\mathbf{\nabla }} \times {\mathbf{G}}\,\text{d}\tau = - \oint\limits_{\text{S}} {{\mathbf{G}} \times \text{d}{\mathbf{s}}} }$.