Numericals / Solved Examples - 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.

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Applications of Derivatives Solved Examples