Vector Calculus Questions and Answers – Gradient of a Function and Conservative Field

This set of Basic Vector Calculus Questions and Answers focuses on “Gradient of a Function and Conservative Field”.

1. Del operator is also known as _________
a) Divergence operator
b) Gradient operator
c) Curl operator
d) Laplacian operator
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2. The gradient of a scalar field V is a vector that represents both magnitude and the direction of the maximum space rate of increase of V.
a) True
b) False
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3. The gradient is taken on a _________
a) tensor
b) vector
c) scalar
d) anything
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4. Find the gradient of a function V if V= xyz.
a) yz a_x + xz a_y + xy a_z
b) yz a_x + xy a_y + xz a_z
c) yx a_x + yz a_y + zx a_z
d) xyz a_x + xy a_y + yz a_z
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5. Find the gradient of V = x² sin(y)cos(z).
a) 2x siny cos z a_x + x² cos(y)cos(z) ay – x² sin(y)sin(z) az
b) 2x siny cos z a_x + x² cos(y)cos(z) ay + x² sin(y)sin(z) az
c) 2x sinz cos y a_x + x² cos(y)cos(z) ay – x² sin(y)sin(z) az
d) x siny cos z a_x + x² cos(y)cos(z) ay – x² sin(y)sin(z) az
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6. Find the gradient of the function W if W = ρzcos(ϕ) if W is in cylindrical coordinates.
a) zcos(ϕ)a_ρ – z sin(ϕ) a_Φ + ρcos(ϕ) a_z
b) zcos(ϕ)a_ρ – sin(ϕ) a_Φ + cos(ϕ) a_z
c) zcos(ϕ)a_ρ + z sin(ϕ) a_Φ + ρcos(ϕ) a_z
d) zcos(ϕ)a_ρ + z sin(ϕ) a_Φ + cos(ϕ) a_z
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7. Find the gradient of A if A = ρ² + z³ + cos(ϕ) + z and A is in cylindrical coordinates.
a) \(2ρz^3 \, a_ρ – \frac{1}{ϕ} sin(ϕ) \, aΦ + 3ρ^2 z^2 \, a_z \)
b) \(2ρz^3 \, a_ρ – \frac{1}{ρ} sin(ϕ) \, aΦ + 3ρ^2 z^2+1 \, a_z \)
c) \(2ρz^3 \, a_ρ – \frac{1}{ϕ} sin(ϕ) \, aΦ + 3ρ^2 z^2+1 \, a_z \)
d) \(2ρz^3 \, a_ρ – \frac{1}{ρ} sin(ϕ) \, aΦ + 3ρ^2 z^2 \, a_z \)
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8. Find gradient of B if B = ϕln(r) + r² ϕ if B is in spherical coordinates.
a) \(\frac{ρ}{r}+ 2rθ \,a_r – r a_θ + \frac{lnr}{rsin(θ)} a_Φ \)
b) \(\frac{ρ}{r}+ 2rϕ \,a_r – r a_θ + \frac{lnr}{rsin(θ)} a_Φ \)
c) \(\frac{ρ}{r}+ 2rθ \,a_r – r^2 a_θ + \frac{lnr}{rsin(θ)} a_Φ \)
d) \( \frac{ρ}{r}+ 2rϕ \,a_r – r^2 a_θ + \frac{lnr}{rsin(θ)} a_Φ \)
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9. Find gradient of B if B = rθϕ if X is in spherical coordinates.
a) \(θϕ \, a_r – ϕ \,a_θ + \frac{θ}{sin(θ)} a_Φ \)
b) \(rθϕ \, a_r – ϕ \,a_θ + r \frac{θ}{sin(θ)} a_Φ \)
c) \(θϕ \, a_r – ϕr \,a_θ + \frac{θ}{sin(θ)} a_Φ \)
d) \(θϕr \, a_r – ϕ \,a_θ + r\frac{θ}{sin(θ)} a_Φ \)
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10. If W = x² y² + xz, the directional derivative \( \frac{dW}{dl} \) in the direction 3 a_x + 4 a_y + 6 a_z at (1,2,0).
a) 5
b) 6
c) 7
d) 8
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11. If W = xy + yz + z, find directional derivative of W at (1,-2,0) in the direction towards the point (3,6,9).
a) -0.6
b) -0.7
c) -0.8
d) -0.9
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12. Electric field E can be written as _________
a) -Gradient of V
b) -Laplacian of V
c) Gradient of V
d) Laplacian of V
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13. Let F = (xy²) a_x + yx² a_y, F is a not a conservative vector.
a) True
b) False
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14. State whether the given equation is a conservative vector.
G = (x³y) a_x + xy³ a_y
a) True
b) False
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15. Find a unit vector normal to the surface of the ellipsoid at (2,2,1) if the ellipsoid is defined as f(x,y,z) = x² + y² + z² – 10.
a) \(\frac{2}{3} a_x + \frac{2}{3} a_y + \frac{1}{3} a_z \)
b) \(\frac{1}{3} a_x + \frac{1}{3} a_y + \frac{1}{3} a_z \)
c) \(\frac{2}{3} a_x + \frac{2}{3} a_y + \frac{2}{3} a_z \)
d) \(\frac{2}{3} a_x + \frac{1}{3} a_y + \frac{1}{3} a_z \)
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