Test 4: Vector Analysis
Instruction: Create a pdf file with all the solutions, upload file on Google Drive and share the link in the form shared at the bottom of the page
Total Marks: 75
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) \(\nabla(\overrightarrow{\boldsymbol{F}} \cdot \overrightarrow{\boldsymbol{G}})\)
(ii) \(\nabla \cdot(\vec{F} \times \vec{G})\)
(iii) \(\nabla \times(\vec{F} \times \vec{G})=\)
(iv) \(\nabla \times(\nabla \times \vec{F})\)
[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]