Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

IIT JAM: Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

The document Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM is a part of the IIT JAM Course Mathematical Models.
All you need of IIT JAM at this link: IIT JAM

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.

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

(a)  The face diagonals Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM are
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

(b) The body diagonals Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM are

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM


Q.2. Calculate the line integral of the function Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM 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 Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM for the loop that goes from a to b along (1) and returns to a along (2)?
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Path (1) consists of two parts. Along the “horizontal” segment dy = dz = 0, so
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
On the “vertical” stretch dx = dz = 0, so
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
By path (1), then,  
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Meanwhile, on path (2) x = y,  dx = dy, and dz = 0, so
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
For the loop that goes out (1) and back (2), then,
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM


Q.3. Find the components of the unit vector nˆ perpendicular to the plane as shown in the figure.
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

The vectors Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM can be defined as
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM


Q.4. Find the line integral of the vector Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM 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.

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

In a Cartesian coordinate system
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM


Q.5. Find the separation vector Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM from the source point (2,8,7) to the field point (4,6,8). Determine its magnitude Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM and construct the unit vectorVector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

The separation vector Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM from the source point (2,8,7) to the field point (4,6,8) is

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Its magnitude Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM and the unit vector
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM


Q.6. (a) Determine whether the force represented byVector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM 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.
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Thus the force Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM is conservative. So work done is independent of paths.
Along line OD , y = x ⇒ dy = dx

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
= [(x2 + y2) dx + 2xy dy] = [(x2 + x2) dx + 2x2dx] = 4x2dx
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM


Q.7. Find the angle between vectors Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM


Q.8. Calculate the surface integral of Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM 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.
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Taking the sides one at a time:
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Evidently the total flux is
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM


Q.9. For given vectors Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
(a) Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
(b) The unit vector along Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
(c) Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
(d) Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
(e) Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

The unit vector along Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM


Q.10. Calculate the volume integral of f = xyz2 over the prism shown in the figure.

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM


Q.11. Find the unit vector perpendicular to both of the vectors Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM and Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

The vectors Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM can be defined as Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAMand Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM


Q.12. Let V = xy2, and take point a to be the origin (0, 0, 0) and b the point (2, 1, 0). Check the fundamental theorem for gradients.

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

V(b) - V(a) = 2


Q.13. (a) 
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
(b) Using the same vectors Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM


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).

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

T = r (cos θ + sin θ cos ϕ) = z + x ⇒ ∇2T = 0
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM


Q.15. Transform the vectorVector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM into Cartesian Coordinates.

x = r sinθ cosϕ , y = r sinθ sinϕ , z = r cosϕ
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM


Q.16. Check the divergence theorem using the function Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM and the unit cube situated at the origin.

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM


Q.17. Transform the vectorVector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM into Cylindrical Coordinates.

x = r cosϕ , y =r sinϕ , z = z
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
⇒ Aϕ = y (-sinϕ) - x (cos ϕ) + z (0) = -r sin2 ϕ - r cos2 ϕ = -r

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM


Q.18. Check the divergence theorem for the function 

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
using the volume of the “ice-crem cone” shown in the figure. 
(The top surface is spherical, with radius R and centered at the origin)

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM


Q.19. Transform the vectorVector Analysis: Assignment Notes | Study Mathematical Models - IIT JAMinto spherical polar Coordinates.

x = r sinθ cosϕ , y = r sinθ sinϕ , z = r cosθ
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
⇒ Ar = 4(sinθ cosϕ) - 2 (sinθ sinϕ) - 4 (cosθ)

⇒ Ar = 2 sinθ [2cosϕ - sinϕ] - 4 (cosθ)
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
⇒ Aθ = 4(cosθ cosϕ) - 2 (cosθ sinϕ) - 4 (- sinθ)

 ⇒ Aθ = 2 cosθ[2cosϕ - sinϕ] + 4 sinθ
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
⇒ Aϕ = 4 (-sinϕ) - 2 (cosϕ) - 4(0) = -4 sinϕ - 2 cosϕ

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM


Q.20. For the vector field Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
(a) Calculate the volume integral of the divergence of Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM out of the region defined by a ≤ x ≤ a, -b ≤ y ≤ b and 0≤ z ≤ c.  
(b) Calculate the flux of Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM out of the region through the surface at z = c. Hence deduce the net flux through the rest of the boundary of the region.

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Thus
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
(b) The flux of Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM out of the region through the surface at z = c is

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM


Q.21. Find a unit vector normal to the surface x2 + 3y2 + 2z2 = 6 at P (2, 0,1).

f = x2 + 3y2 + 2z2 - 6 = 0
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM


Q.22. Consider a vector Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
(a) Calculate the line integral Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM from point P→O along the path P→Q→R→O as shown in the figure.
(b) Using Stokes’s theorem appropriately, calculate Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM for the same path P→Q→R→O.
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

The line integral Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM from point P → O is
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Along line PQ , y = 1 ⇒ dy = 0
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Along line QR , x = 1 ⇒ dx = 0
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Along line RO , y = 0 ⇒ dy = 0
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM


Q.23. Find the unit vector normal to the curve y = x2 at the point (2, 4, 1).

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM


Q.24. How much work is done when an object moves from O → P → Q → R → O in a force field given by Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM along the rectangular path shown. Find the answer by evaluating the line integral and also by using the Stokes’ theorem. 

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Using the Stokes’ theorem
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM


Q.25. Find the unit vector normal to the surface xy3z2 = 4 at a point (-1, -1, 2).

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM


Q.26. (a) Consider a constant vector field Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAMFind any one of the many possible vectors Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM for which Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
(b) Using Stoke’s theorem, evaluate the flux associated with the field Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAMthrough the curved hemispherical surface defined by x2 + y2 + z2 = r2, z > 0.

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
We have to take line integral around circle x2 + y2 = r2 in z = 0 plane. Let use cylindrical coordinate and use x = r cosϕ , y = r sinϕ ⇒ dy = r cosϕdϕ.
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Q.27. Calculate the divergence of the following vector functions:

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM


Q.28. Compute the line integral of Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAMalong the triangular path shown in figure. Check your answer using Stoke’s theorem.
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM


Q.29. Calculate the curls of the following vector functions:
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM


Q.30. Check Stoke’s theorem for the functionVector Analysis: Assignment Notes | Study Mathematical Models - IIT JAMusing the triangular surface shown in figure below.
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM


Q.31. Calculate the Laplacian of the following functions: 
(a) f(x, y, z) = x2 + 2xy + 3z + 4 
(b) f(x, y, z) = sin(x) sin(y) sin(z) 
(c) f(x, y, z) = e-5x sin4y cos3z 
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM

Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM
Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM 

The document Vector Analysis: Assignment Notes | Study Mathematical Models - IIT JAM is a part of the IIT JAM Course Mathematical Models.
All you need of IIT JAM at this link: IIT JAM
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