Solved Examples/Numericals - Gradient Of A Scalar.

Q. 1 Find the gradient of these scalar fields:
a) U = 4xz₂ + 3yz
b) H = r²cosθ cosφ
Ans:
a)

=>∇ U = 4z² a_x + 3z a_y + (8xz +3y) a_z

b)

=>∇ H = 2r cosθ cosφ a_r + r sinθ cosφ a_θ + rcosθ sinφ a_φ 


Q.2 If V_{(x, y, z)} = 3x²y –y²z², find ∇ V and |∇ V| at the point (1, 2, -1).
Ans:

∇ V = 6xy a_x + (3x² – 2yz²) a_y + (-2y²z) a_z

At the point (1, 2, -1)

∇ V = 6(1)(2) a_x + [3(12) – 2(2)(-1)²] a_y – 2(2)² (-1) a_z
=>∇ V = 12a_x - a_y + 8a_z

|∇ V| _{(1, 2, -1)} = |12a_x - a_y + 8a_z | = (209)^1/2


Q.3 Given φ = xy +yz +xz, find gradient φ at point (1, 2, 3) and thedirectional derivative of φ
Ans:
∇ φ = (y + z)a_x + (x + z)a_y + (y + z)a_z

At point (1, 2, 3)

∇ φ = 5a_x + 4a_y + 3a_z

The directional derivative is given as:

dφ/dl = ∇ φ . a_l
= (5, 4, 3) . [(3, 4, 4) – (1, 2, 3)] / 3
= [(5, 4, 3) . (2, 2, 1)] / 3 = 7


Q.4 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.
Ans:

∂V / ∂x = y² -2xyz³

∂V / ∂y = 3 + 2xy – x²z³

∂V / ∂y = 4z³ – 3x²yz²

V = ∫( y² – 2xyz³) dx = xy² – x²yz³ + f(y, z)

V = ∫( 3 + 2xy – x²z³) dy = 3y + xy² – x²yz³ + g(x, z)

V = ∫( 4z³ – 3x²yz²) dz = z⁴ – x²yz³ + h(x, y)

Comparing the above 3 equations, we have
f (x, y) = 3y + z⁴ + c

Hence,

V = xy² – x²yz³ + 3y + z⁴ + c

Since V_{(0, 0, 0)} = -2

V = xy² – x²yz³ + 3y + z⁴ – 2


Q.5 Find the unit normal vector of the surface x² + y² + z² = 14 at (-1, 3, 2) ?
Ans:
Here V _{(x, y, z)} = x² + y² + z²

∇ V = 2x a_x + 2y a_y + 2z a_z

At point (-1, 3, 2)

∇ V = -2a_x + 6a_y + 4a_z

|∇ V| = 2(14)^1/2

Unit normal vector
a_n = ∇ V / | ∇ V |
= - a_x + 3a_y + 2a_z / (14)^1/2


Q.6 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?

Ans:

∇ T = 2xa_x + 2ya_y - a_z

At point (1, 1, 2)

∇ T = 2a_x + 2a_y - a_z

The mosquito should move in the direction of 2a_x + 2a_y - a_z.




ALSO READ:

- Line , Surface and Volume Intergral.

- 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).


Your suggestions and comments are welcome in this section. If you want to share something or if you have some stuff of your own, please do post them in the comments section.