Q.1. (a) Find the angle between the face diagonals of a cube of unit length.
(b) Find the angle between the body diagonals of a cube of unit length.
(a) The face diagonals are
(b) The body diagonals are
Q.2. Calculate the line integral of the function from the point a = (1, 1, 0) to the point b = (2, 2, 0) along the paths (1) and (2) as shown in figure. What is for the loop that goes from a to b along (1) and returns to a along (2)?
Path (1) consists of two parts. Along the “horizontal” segment dy = dz = 0, so
On the “vertical” stretch dx = dz = 0, so
By path (1), then,
Meanwhile, on path (2) x = y, dx = dy, and dz = 0, so
For the loop that goes out (1) and back (2), then,
Q.3. Find the components of the unit vector nˆ perpendicular to the plane as shown in the figure.
The vectors can be defined as
Q.4. Find the line integral of the vector around a square of side ‘b’ which has a corner at the origin, one side on the x axis and the other side on the y axis.
In a Cartesian coordinate system
Q.5. Find the separation vector from the source point (2,8,7) to the field point (4,6,8). Determine its magnitude and construct the unit vector
The separation vector from the source point (2,8,7) to the field point (4,6,8) is
Its magnitude and the unit vector
Q.6. (a) Determine whether the force represented by is conservative or not. Here k = 1Nm^{2}.
(b) Calculate the work done by this force in moving a particle from the origin O (0, 0, 0) to the point D(1, 1, 0) on the z = 0 plane along the paths OABD and OD as shown in the figure, where the coordinates are measured in meters.
Thus the force is conservative. So work done is independent of paths.
Along line OD , y = x ⇒ dy = dx
= [(x^{2} + y^{2}) dx + 2xy dy] = [(x^{2} + x^{2}) dx + 2x^{2}dx] = 4x^{2}dx
Q.7. Find the angle between vectors
Q.8. Calculate the surface integral of over five sides (excluding the bottom) of the cubical box (side 2) as shown in figure. Let “upward and outward” be the positive direction, as indicated by the arrows.
Taking the sides one at a time:
Evidently the total flux is
Q.9. For given vectors
(a)
(b) The unit vector along
(c)
(d)
(e)
The unit vector along
Q.10. Calculate the volume integral of f = xyz^{2} over the prism shown in the figure.
Q.11. Find the unit vector perpendicular to both of the vectors and
The vectors can be defined as and
Q.12. Let V = xy^{2}, and take point a to be the origin (0, 0, 0) and b the point (2, 1, 0). Check the fundamental theorem for gradients.
V(b)  V(a) = 2
Q.13. (a)
(b) Using the same vectors
Q.14. Compute the gradient and Laplacian of the function T = r (cos θ + sin θ cos θ). Check the Laplacian by converting T to Cartesian coordinates. Test the gradient theorem for this function, using the path shown in figure, from (0, 0, 0) to (0, 0, 2).
T = r (cos θ + sin θ cos ϕ) = z + x ⇒ ∇^{2}T = 0
Q.15. Transform the vector into Cartesian Coordinates.
x = r sinθ cosϕ , y = r sinθ sinϕ , z = r cosϕ
Q.16. Check the divergence theorem using the function and the unit cube situated at the origin.
Q.17. Transform the vector into Cylindrical Coordinates.
x = r cosϕ , y =r sinϕ , z = z
⇒ Aϕ = y (sinϕ)  x (cos ϕ) + z (0) = r sin^{2} ϕ  r cos^{2} ϕ = r
Q.18. Check the divergence theorem for the function
using the volume of the “icecrem cone” shown in the figure.
(The top surface is spherical, with radius R and centered at the origin)
Q.19. Transform the vectorinto spherical polar Coordinates.
x = r sinθ cosϕ , y = r sinθ sinϕ , z = r cosθ
⇒ A_{r} = 4(sinθ cosϕ)  2 (sinθ sinϕ)  4 (cosθ)⇒ A_{r} = 2 sinθ [2cosϕ  sinϕ]  4 (cosθ)
⇒ A_{θ} = 4(cosθ cosϕ)  2 (cosθ sinϕ)  4 ( sinθ)⇒ A_{θ} = 2 cosθ[2cosϕ  sinϕ] + 4 sinθ
⇒ A_{ϕ} = 4 (sinϕ)  2 (cosϕ)  4(0) = 4 sinϕ  2 cosϕ
Q.20. For the vector field
(a) Calculate the volume integral of the divergence of out of the region defined by a ≤ x ≤ a, b ≤ y ≤ b and 0≤ z ≤ c.
(b) Calculate the flux of out of the region through the surface at z = c. Hence deduce the net flux through the rest of the boundary of the region.
Thus
(b) The flux of out of the region through the surface at z = c is
Q.21. Find a unit vector normal to the surface x^{2} + 3y^{2} + 2z^{2} = 6 at P (2, 0,1).
f = x^{2} + 3y^{2} + 2z^{2}  6 = 0
Q.22. Consider a vector
(a) Calculate the line integral from point P→O along the path P→Q→R→O as shown in the figure.
(b) Using Stokes’s theorem appropriately, calculate for the same path P→Q→R→O.
The line integral from point P → O is
Along line PQ , y = 1 ⇒ dy = 0
Along line QR , x = 1 ⇒ dx = 0Along line RO , y = 0 ⇒ dy = 0
Q.23. Find the unit vector normal to the curve y = x^{2} at the point (2, 4, 1).
Q.24. How much work is done when an object moves from O → P → Q → R → O in a force field given by along the rectangular path shown. Find the answer by evaluating the line integral and also by using the Stokes’ theorem.
Using the Stokes’ theorem
Q.25. Find the unit vector normal to the surface xy^{3}z^{2} = 4 at a point (1, 1, 2).
Q.26. (a) Consider a constant vector field Find any one of the many possible vectors for which
(b) Using Stoke’s theorem, evaluate the flux associated with the field through the curved hemispherical surface defined by x^{2} + y^{2} + z^{2} = r^{2}, z > 0.
We have to take line integral around circle x^{2} + y^{2} = r^{2} in z = 0 plane. Let use cylindrical coordinate and use x = r cosϕ , y = r sinϕ ⇒ dy = r cosϕdϕ.
Q.27. Calculate the divergence of the following vector functions:
Q.28. Compute the line integral of along the triangular path shown in figure. Check your answer using Stoke’s theorem.
Q.29. Calculate the curls of the following vector functions:
Q.30. Check Stoke’s theorem for the functionusing the triangular surface shown in figure below.
Q.31. Calculate the Laplacian of the following functions:
(a) f(x, y, z) = x^{2} + 2xy + 3z + 4
(b) f(x, y, z) = sin(x) sin(y) sin(z)
(c) f(x, y, z) = e^{5x} sin4y cos3z
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