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Vector Analysis: Assignment Notes | Study Mathematical Models - Physics

Document Description: Vector Analysis: Assignment for Physics 2022 is part of Mathematical Models preparation. The notes and questions for Vector Analysis: Assignment have been prepared according to the Physics exam syllabus. Information about Vector Analysis: Assignment covers topics like and Vector Analysis: Assignment Example, for Physics 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Vector Analysis: Assignment.

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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 - Physics

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

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

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics


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

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


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 - Physics

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


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

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


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

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

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


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

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics

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

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


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

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics


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

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


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

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics

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

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics
Vector Analysis: Assignment Notes | Study Mathematical Models - Physics


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

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics


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

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


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 - Physics

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics

V(b) - V(a) = 2


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

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics
Vector Analysis: Assignment Notes | Study Mathematical Models - Physics

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics


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 - Physics

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


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

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


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

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics


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

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

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics
Vector Analysis: Assignment Notes | Study Mathematical Models - Physics


Q.18. Check the divergence theorem for the function 

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics
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 - Physics

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics


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

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

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

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

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics


Q.20. For the vector field Vector Analysis: Assignment Notes | Study Mathematical Models - Physics
(a) Calculate the volume integral of the divergence of Vector Analysis: Assignment Notes | Study Mathematical Models - Physics 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 - Physics 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 - Physics
Thus
Vector Analysis: Assignment Notes | Study Mathematical Models - Physics
Vector Analysis: Assignment Notes | Study Mathematical Models - Physics

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

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics
Vector Analysis: Assignment Notes | Study Mathematical Models - Physics
Vector Analysis: Assignment Notes | Study Mathematical Models - Physics


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 - Physics

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics


Q.22. Consider a vector Vector Analysis: Assignment Notes | Study Mathematical Models - Physics
(a) Calculate the line integral Vector Analysis: Assignment Notes | Study Mathematical Models - Physics 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 - Physics for the same path P→Q→R→O.
Vector Analysis: Assignment Notes | Study Mathematical Models - Physics

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

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

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

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics

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

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics


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 - Physics
Vector Analysis: Assignment Notes | Study Mathematical Models - Physics


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 - Physics 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 - Physics

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


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 - Physics


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

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics
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 - Physics

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

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics
Vector Analysis: Assignment Notes | Study Mathematical Models - Physics

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics
Vector Analysis: Assignment Notes | Study Mathematical Models - Physics
Vector Analysis: Assignment Notes | Study Mathematical Models - Physics
Vector Analysis: Assignment Notes | Study Mathematical Models - Physics

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics
Vector Analysis: Assignment Notes | Study Mathematical Models - Physics
Vector Analysis: Assignment Notes | Study Mathematical Models - Physics
Vector Analysis: Assignment Notes | Study Mathematical Models - Physics
Vector Analysis: Assignment Notes | Study Mathematical Models - Physics


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

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics


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

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics
Vector Analysis: Assignment Notes | Study Mathematical Models - Physics
Vector Analysis: Assignment Notes | Study Mathematical Models - Physics
Vector Analysis: Assignment Notes | Study Mathematical Models - Physics


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

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics
Vector Analysis: Assignment Notes | Study Mathematical Models - Physics


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 - Physics

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics

Vector Analysis: Assignment Notes | Study Mathematical Models - Physics
Vector Analysis: Assignment Notes | Study Mathematical Models - Physics 

The document Vector Analysis: Assignment Notes | Study Mathematical Models - Physics is a part of the Physics Course Mathematical Models.
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