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 a_ρ+ z sinφ a_z around the edge L of the wedge defined by 0 < ρ < 2, 0 < φ < 60^o, z = 0 as shown.








SOLUTION/ANSWER




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


Q.3 Given that ρ_s = x² + xy, calculate ∫_s ρ_sds 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)a_x + (3 + 2xy – x²z³)a_y + (4z³ – 3x²yz²)a_z 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 a_x + 4xy a_y + y a_z at point (1, -2, 3)
b) A = ρzsinφ a_ρ + 3ρz² cosφ a_φ at (5, π / 2, 1)
c) A = 2r cosθ cosφ a_r + r^1/2 a_φ   at (1, π / 6, π / 3)    SOLUTION/ANSWER


Q.11 Determine the flux of D = ρ² cos²φ a_ρ + zsin φ a_φ 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 a_x + 4xy a_y + y a_z
b) A = ρz sinφ a_ρ + 3ρz²cosφ a_φ             SOLUTION/ANSWER


Q.14 Show that A = (y + z cos xz) a_x + x a_y + xcosxz a_z 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


\