1. Write the expressions for line element and volume element in cylindrical coordinates.

2. Write the expression for divergence of a vector inorthogonalcurvilinear coordinates.

3. Write the expression for Laplacian of a scalar in orthogonal curvilinear coordinates.

4. Derive the expression for curl of a vector in orthogonal curvilinear coordinates.

5. Describe the spherical coordinate system with necessary diagram to derive the expression for divergence of a vector in spherical coordinates.

6. Write the values of scale factors h₁, h₂ and h₃ of curvilinear coordinate system in spherical polar coordinate system.

7. Give the expression for gradient of a scalar function φ in curvilinear coordinate system.

8. Write the expression for Laplacian operator in spherical polar coordinate system.

9. Write the expression for line and volume elements in cylindrical coordinate system.

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10. Derive an expression for the divergence of a vector in curvilinear coordinate system.

11. Write the expression for the gradient of a scalar function in cylindrical coordinate system. Prove that cylindrical coordinate system is orthogonal.

12. Write the expression for line element spherical polar coordinate system.

13. Write the expression for volume element in cylindrical coordinate system.

14. Write the expression for gradient of a scalar function in cylindrical coordinate system.

15. Find the expression for

∇²φ

in orthogonal curvilinear coordinate system.

16. If

u = 2x + 3,   v = y − 4,   w = z + 2

and

r = xi + yj + zk

be the position vector, i.e.,

r = xi + yj + zk,

then show that

∂r ∂u ,   ∂r ∂v and ∂r ∂w

are mutually orthogonal.

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17. Use the Newton-Raphson method to find a root of the equation

x³ − 2x − 5 = 0.

18. Evaluate

I = ∫₀¹ 1 1 + x dx

correct to three decimal places using Simpson rule with

h = 0.125.

19. hat do you mean by orthogonal coordinate system?

20. write down the transformation equations from Cartesian coordinates to cylindrical coordinates.

21. Write down the expressions of

∇φ

and

∇ · A

in spherical polar coordinates, where φ is a scalar function and A is a vector.

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22. Find the expressions for line element and volume element in orthogonal curvilinear coordinates. Also find out the components of the vector

A = 2yi − 3j + 2zk

in cylindrical polar coordinate system.

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23. Deduce the expression for divergence, curl and Laplacian operator in general orthogonal coordinate system.

24. Find the expression for element of arc length in curvilinear coordinates.

25. Write down the expression for Laplacian in orthogonal curvilinear coordinates.

26. Write the transformation equations between rectangular coordinates and spherical polar coordinates.

27. Explain the cylindrical coordinate system. Find the expression for length and volume elements in cylindrical coordinate system. Also find the expression for gradient and divergence in this system.

28. Explain curvilinear coordinates.
Define orthogonal curvilinear coordinates.
Find the expression for unit vectors in curvilinear coordinates.

29. Show that in spherical polar coordinates the unit vectors are mutually perpendicular to each other.