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Test 5: Dynamics

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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]


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