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Exercises - Set 1

Topic: Mathematical Background

Electromagnetics I Prof. Dr. Gökhan Çınar Eskişehir Osmangazi University Electrical and Electronics Engineering Department

1. Using the differential length dl, find the length of each of the following curves: (a) ρ = 3, π/4 < φ < π/2, z =constant (b) r = 1, θ = π/6, 0 < φ < π/3, (c) r = 4, π/6 < θ < π/2, φ =constant. 2. Calculate the areas of the following surfaces using the differential surface area dS. (a) ρ = 3, 0 < z < 5, π/3 < φ < π/2 (b) z = 1, 1 < ρ < 3, 0 < φ < π/4 (c) r = 10, π/4 < θ < 2π/3, 0 < φ < 2π (d) 0 < r < 4, π/3 < θ < π/2, φ =constant 3. Prove that the surface area of a sphere with a radius of a is 4πa2 . 4. Use the differential volume dv to determine the volumes of the following regions. (a) 0 < x < 1, 1 < y < 2, −3 < z < 3 (b) 2 < ρ < 5, π/3 < φ < π, −1 < z < 4 (c) 1 < r < 3, π/2 < θ < 2π/3, π/6 < φ < π/2 ∫ 5. Given that f (x, y) = x2 + xy, calculate S f dS over the region y ≤ x2 , 0 < x < 1. ∫ ⃗ = x2⃗ax + y2⃗ay , evaluate A ⃗ · d⃗l where L is along the curve y = x2 from (0, 0) to 6. Given that A L (1, 1). √ 7. For⃗r = x⃗ex + y⃗ey + z⃗ez and r = x2 + y2 + z2 , show that ( ) ⃗r 1 ∇ = − 3. r r 8. Prove that the identity ∇ × ∇f = 0 is valid for any scalar function f (x, y, z). ⃗ = 0 is valid for any vector function A ⃗ (x, y, z). 9. Prove that the identity ∇ · ∇ × A 10. Calculate gradient of f (ρ, φ, z) = 2ρ (z2 + 1) cos φ. 11. Calculate gradient of f (r, θ, φ) = r2 cos θ cos φ. ⃗ = exy⃗ax + sin xy⃗ay + cos2 xz⃗az . 12. Calculate the divergence of A ⃗ = ρz2 cos φ⃗aρ + z2 sin2 φ⃗az . 13. Calculate the divergence of A

Prof. Dr. Gökhan Çınar

⃗ = r cos θ⃗ar − 1 sin θ⃗aθ + 2r2 sin θ⃗aφ . 14. Calculate the divergence of A r ⃗ = exy⃗ax + sin xy⃗ay + cos2 xz⃗az . 15. Calculate the curl of A ⃗ = ρz2 cos φ⃗aρ + z2 sin2 φ⃗az . 16. Calculate the curl of A ⃗ = r cos θ⃗ar − 1 sin θ⃗aθ + 2r2 sin θ⃗aφ . 17. Calculate the curl of A r 18. Calculate the Laplacian of f = x3 y2 exz at (1, −1, 1). 19. Calculate the Laplacian of f = ρ2 z (cos φ + sin φ) at (5, π/6, −2). 20. Calculate the Laplacian of f = e−r sin θ cos φ at (1, π/3, π/6). 21. Calculate total mass of a spherical ball of radius a with a density of 2r2 kg/m3 . 22. Verify the fundamental theorem for gradients with f (x, y) = xy2 by considering the curves in the figure and the points ⃗a = (0, 0) and b = (2, 1).

⃗ = y2⃗ex + (2xy + z2 )⃗ey + 2yz⃗ez and the cube given in the figure. 23. Verify Gauss’s theorem using A

⃗ = (2xz + 3y2 )⃗ey + 4yz2⃗ez and the square given in the figure 24. Verify Stokes’ theorem using A

⃗ = 4r2⃗er and a sphere of radius a centered at the origin. 25. Verify Gauss’s theorem using A ⃗ = 2ρ⃗eφ and the disk of radius a, located on the xy−plane with 26. Verify Stokes’ theorem using A its center at the origin.