Test 4: Vector Analysis

1) Define the following terms with an example of each:
i) Scalar and Vector Point functions
ii) Level Surface

[4M]

2.(i) Define gradient. What does it physically represent?
2.(ii) Define directional derivative. Write the expression to evaluate it.
2.(iii) Write the equations of tangent plane and normal to a given surface.
2.(iv) Write the formula for calculating the angle between two surfaces at a common point.

[12M]

3.(i) Define divergence. What does it physically represent?
3.(ii) What do you mean by a solenoidal vector field?

[5M]

4) Expand below identities:
(i) $\mathrm{\nabla }(\overrightarrow{\mathit{F}}\cdot \overrightarrow{\mathit{G}})$
(ii) $\mathrm{\nabla }\cdot (\overrightarrow{F}\times \overrightarrow{G})$
(iii) $\mathrm{\nabla }\times (\overrightarrow{F}\times \overrightarrow{G})=$
(iv) $\mathrm{\nabla }\times (\mathrm{\nabla }\times \overrightarrow{F})$

[12M]

5.(i) Define curl. What does it represent?
5.(ii) Define conservative vector field.
5.(iii) Define irrotational vector field.

[9M]

6) State Green’s theorem for two scalar functions. Also, write the vector form of Green’s theorem.

[8M]

7) State Stokes’ Theorem.

[8M]

8) State Gauss divergence theorem.

[8M]

9) State and prove Green’s three identities.

[9M]