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Vector Analysis: Assignment | Mathematical Methods - Physics PDF Download

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 | Mathematical Methods - Physics

(a)  The face diagonals Vector Analysis: Assignment | Mathematical Methods - Physics are
Vector Analysis: Assignment | Mathematical Methods - Physics
Vector Analysis: Assignment | Mathematical Methods - Physics

(b) The body diagonals Vector Analysis: Assignment | Mathematical Methods - Physics are

Vector Analysis: Assignment | Mathematical Methods - Physics

Vector Analysis: Assignment | Mathematical Methods - Physics


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

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


Q.3. Find the components of the unit vector nˆ perpendicular to the plane as shown in the figure.
Vector Analysis: Assignment | Mathematical Methods - Physics

The vectors Vector Analysis: Assignment | Mathematical Methods - Physics can be defined as
Vector Analysis: Assignment | Mathematical Methods - Physics
Vector Analysis: Assignment | Mathematical Methods - Physics


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

In a Cartesian coordinate system
Vector Analysis: Assignment | Mathematical Methods - Physics
Vector Analysis: Assignment | Mathematical Methods - Physics
Vector Analysis: Assignment | Mathematical Methods - Physics
Vector Analysis: Assignment | Mathematical Methods - Physics
Vector Analysis: Assignment | Mathematical Methods - Physics
Vector Analysis: Assignment | Mathematical Methods - Physics


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

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

Vector Analysis: Assignment | Mathematical Methods - Physics
Its magnitude Vector Analysis: Assignment | Mathematical Methods - Physics and the unit vector
Vector Analysis: Assignment | Mathematical Methods - Physics


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

Vector Analysis: Assignment | Mathematical Methods - Physics

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

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


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

Vector Analysis: Assignment | Mathematical Methods - Physics


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

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


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

Vector Analysis: Assignment | Mathematical Methods - Physics

The unit vector along Vector Analysis: Assignment | Mathematical Methods - Physics

Vector Analysis: Assignment | Mathematical Methods - Physics
Vector Analysis: Assignment | Mathematical Methods - Physics


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

Vector Analysis: Assignment | Mathematical Methods - Physics

Vector Analysis: Assignment | Mathematical Methods - Physics


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

The vectors Vector Analysis: Assignment | Mathematical Methods - Physics can be defined as Vector Analysis: Assignment | Mathematical Methods - Physicsand Vector Analysis: Assignment | Mathematical Methods - Physics
Vector Analysis: Assignment | Mathematical Methods - 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 | Mathematical Methods - Physics

Vector Analysis: Assignment | Mathematical Methods - Physics

Vector Analysis: Assignment | Mathematical Methods - Physics

V(b) - V(a) = 2


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

Vector Analysis: Assignment | Mathematical Methods - Physics
Vector Analysis: Assignment | Mathematical Methods - Physics

Vector Analysis: Assignment | Mathematical Methods - 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 | Mathematical Methods - Physics

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


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

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


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

Vector Analysis: Assignment | Mathematical Methods - Physics


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

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

Vector Analysis: Assignment | Mathematical Methods - Physics
Vector Analysis: Assignment | Mathematical Methods - Physics


Q.18. Check the divergence theorem for the function 

Vector Analysis: Assignment | Mathematical Methods - 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 | Mathematical Methods - Physics

Vector Analysis: Assignment | Mathematical Methods - Physics


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

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

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

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

Vector Analysis: Assignment | Mathematical Methods - Physics


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

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

Vector Analysis: Assignment | Mathematical Methods - Physics
Vector Analysis: Assignment | Mathematical Methods - Physics
Vector Analysis: Assignment | Mathematical Methods - 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 | Mathematical Methods - Physics

Vector Analysis: Assignment | Mathematical Methods - Physics


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

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

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

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

Vector Analysis: Assignment | Mathematical Methods - Physics

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

Vector Analysis: Assignment | Mathematical Methods - Physics


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

Vector Analysis: Assignment | Mathematical Methods - Physics
Vector Analysis: Assignment | Mathematical Methods - 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 | Mathematical Methods - Physics along the rectangular path shown. Find the answer by evaluating the line integral and also by using the Stokes’ theorem. 

Vector Analysis: Assignment | Mathematical Methods - Physics

Vector Analysis: Assignment | Mathematical Methods - Physics
Vector Analysis: Assignment | Mathematical Methods - Physics
Using the Stokes’ theorem
Vector Analysis: Assignment | Mathematical Methods - Physics


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

Vector Analysis: Assignment | Mathematical Methods - Physics


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

Vector Analysis: Assignment | Mathematical Methods - Physics

Vector Analysis: Assignment | Mathematical Methods - 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 | Mathematical Methods - Physics

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

Vector Analysis: Assignment | Mathematical Methods - Physics

Vector Analysis: Assignment | Mathematical Methods - Physics
Vector Analysis: Assignment | Mathematical Methods - Physics

Vector Analysis: Assignment | Mathematical Methods - Physics
Vector Analysis: Assignment | Mathematical Methods - Physics
Vector Analysis: Assignment | Mathematical Methods - Physics
Vector Analysis: Assignment | Mathematical Methods - Physics

Vector Analysis: Assignment | Mathematical Methods - Physics
Vector Analysis: Assignment | Mathematical Methods - Physics
Vector Analysis: Assignment | Mathematical Methods - Physics
Vector Analysis: Assignment | Mathematical Methods - Physics
Vector Analysis: Assignment | Mathematical Methods - Physics


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

Vector Analysis: Assignment | Mathematical Methods - Physics


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

Vector Analysis: Assignment | Mathematical Methods - Physics

Vector Analysis: Assignment | Mathematical Methods - Physics

Vector Analysis: Assignment | Mathematical Methods - Physics
Vector Analysis: Assignment | Mathematical Methods - Physics
Vector Analysis: Assignment | Mathematical Methods - Physics
Vector Analysis: Assignment | Mathematical Methods - Physics


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

Vector Analysis: Assignment | Mathematical Methods - Physics
Vector Analysis: Assignment | Mathematical Methods - 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 | Mathematical Methods - Physics

Vector Analysis: Assignment | Mathematical Methods - Physics

Vector Analysis: Assignment | Mathematical Methods - Physics
Vector Analysis: Assignment | Mathematical Methods - Physics 

The document Vector Analysis: Assignment | Mathematical Methods - Physics is a part of the Physics Course Mathematical Methods.
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FAQs on Vector Analysis: Assignment - Mathematical Methods - Physics

1. What is vector analysis?
Ans. Vector analysis is a branch of mathematics that deals with the study of vectors and their properties. It involves the analysis of vectors in terms of their magnitude, direction, and spatial relationships. This mathematical tool is widely used in various fields such as physics, engineering, computer science, and economics to analyze and solve problems involving quantities that have both magnitude and direction.
2. How is vector analysis used in IIT JAM?
Ans. In IIT JAM, vector analysis is an important topic under the Mathematics section. Questions related to vectors are frequently asked in the exam to test the candidates' understanding of vector operations, vector algebra, vector calculus, and applications of vectors. A strong grasp of vector analysis is essential for solving problems in other topics such as calculus, differential equations, and physics.
3. What are the key concepts covered in vector analysis for IIT JAM?
Ans. Vector analysis for IIT JAM covers various key concepts, including vector addition, subtraction, and scalar multiplication. Other important topics include dot product, cross product, vector equations of lines and planes, vector differentiation and integration, divergence, curl, and gradient of vectors. Understanding these concepts and their applications is crucial for performing well in the exam.
4. How can I improve my understanding of vector analysis for IIT JAM?
Ans. To improve your understanding of vector analysis for IIT JAM, it is recommended to start by thoroughly studying the fundamentals of vector operations and vector algebra. Practice solving a variety of vector-related problems to develop your problem-solving skills. Additionally, refer to textbooks, online resources, and video lectures specifically designed for IIT JAM preparation. Taking mock tests and solving previous years' question papers will also help in gaining familiarity with the exam pattern and boosting your confidence.
5. Are there any real-life applications of vector analysis?
Ans. Yes, vector analysis has numerous real-life applications. It is widely used in physics to study the motion of objects, forces acting on them, and electromagnetic fields. In engineering, vector analysis is utilized in fields such as structural analysis, fluid mechanics, and electrical circuits. It is also employed in computer graphics, robotics, navigation systems, and optimization problems. Understanding vector analysis not only helps in solving theoretical problems but also equips individuals with practical problem-solving skills in various disciplines.
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