Vector Calculus


This section deals with integration and differentiation of vectors. This section helps you understand how to use coordinate system and the different jargon's used while understanding Electrostatics, Magnetostatics, Waves & Applications.

 Vector calculus (or vector analysis) is a branch of mathematics concerned with differentiation and integration of vector fields, primarily in 3 dimensional space. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields and fluid flow.

Following links will throw more light on the sub-topics:


- Line , Surface and Volume Integral.

- Del Operator - Definition and Significance.

- Gradient Of a Scalar (∇ V).

- Numericals / Solved Examples - Gradient Of a Scalar.


- Divergence Of a Vector ( ∇ . A ).

- Numericals / Solved Examples - Divergence Of a Vector.

- Curl Of a Vector ( ∇ x A).

- Laplacian Of a Scalar ( ∇2 V).


SOLVED NUMERICALS ON VECTOR CALCULUS:

Q.1 Calculate the circulation of A = ρ cosφ aρ+ z sinφ az around the edge L of the wedge defined by 0 < ρ < 2, 0 < φ < 60o, z = 0 as shown.








SOLUTION/ANSWER




Q.2 Given that H = x2 ax + y2 ay, evaluate ∫L H .dl where L is along the curve y =x2 from (0, 0) to (1, 1).                    SOLUTION/ANSWER


Q.3 Given that ρs = x2 + xy, calculate ∫s ρsds over the region y ≤ x2, 0< x< 1.               SOLUTION/ANSWER


Q.4 Find the gradient of these scalar fields:
a) U = 4xz2 + 3yz
b) H = r2cosθ cosφ                SOLUTION/ANSWER


Q.5 If V(x, y, z) = 3x2y –y2z2, find ∇ V and |∇ V| at the point (1, 2, -1).   SOLUTION/ANSWER


Q.6 Given φ = xy +yz +xz, find gradient φ at point (1, 2, 3) and the directional derivative of φ.     SOLUTION/ANSWER


Q.7 Find V(x, y, z) if grad V = (y2 – 2xyz3)ax + (3 + 2xy – x2z3)ay + (4z3 – 3x2yz2)az and V(0, 0, 0) = -2.      SOLUTION/ANSWER


Q.8 Find the unit normal vector of the surface x2 + y2 + z2 = 14 at (-1, 3, 2) ?     SOLUTION/ANSWER


Q.9 The temperature in an auditorium is given by T = x2 + y2 – z. A mosquito located at (1, 1, 2) in the auditorium desires to fly in such a direction that it will get warm as soon as possible. In what direction must it fly?      SOLUTION/ANSWER


Q.10 Determine the divergence of the following vector fields and evaluate them at the specified points:

a) A = yz ax + 4xy ay + y az at point (1, -2, 3)
b) A = ρzsinφ aρ + 3ρz2 cosφ aφ at (5, π / 2, 1)
c) A = 2r cosθ cosφ ar + r1/2 aφ   at (1, π / 6, π / 3)    SOLUTION/ANSWER


Q.11 Determine the flux of D = ρ2 cos2φ aρ + zsin φ aφ over the closed surface of the cylinder, 0 < z < 1, ρ =4. Verify the divergence theorem for this case.       SOLUTION/ANSWER


Q.13 Determine the curl of each of the vector field:
a) A = yz ax + 4xy ay + y az
b) A = ρz sinφ aρ + 3ρz2cosφ aφ             SOLUTION/ANSWER


Q.14 Show that A = (y + z cos xz) ax + x ay + xcosxz az is conservative, without computing any integrals?                                    SOLUTION/ANSWER


Q.15 Determine the Laplacian of the scalar fields:
a) V = x2y + xyz
b) V = ρz sinφ + z2cos2 φ + ρ2                   SOLUTION/ANSWER


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