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
View Answer
Answer: b
Explanation: This differential operator is not a vector itself but when it operates on a scalar function, for example, a vector ensues.
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
View Answer
Answer: a
Explanation: A gradient operates on a scalar only and gives a vector as a result. This vector has a magnitude and direction. The gradient is found by finding the speed that is by taking the partial differentiation.
3. The gradient is taken on a _________
a) tensor
b) vector
c) scalar
d) anything
View Answer
Answer: c
Explanation: Gradient is taken only on a scalar field. After taking gradient of a scalar field it becomes a vector. It is found by taking the partial differentiation.
4. Find the gradient of a function V if V= xyz.
a) yz ax + xz ay + xy az
b) yz ax + xy ay + xz az
c) yx ax + yz ay + zx az
d) xyz ax + xy ay + yz az
View Answer
Answer: a
Explanation: V = xyz
Gradient of (V = frac{∂V}{∂x} a_x + frac{∂V}{∂y}a_y + frac{∂V}{∂z}a_z )
= yz ax + xz ay + xy az
5. Find the gradient of V = x2 sin(y)cos(z).
a) 2x siny cos z ax + x2 cos(y)cos(z) ay – x2 sin(y)sin(z) az
b) 2x siny cos z ax + x2 cos(y)cos(z) ay + x2 sin(y)sin(z) az
c) 2x sinz cos y ax + x2 cos(y)cos(z) ay – x2 sin(y)sin(z) az
d) x siny cos z ax + x2 cos(y)cos(z) ay – x2 sin(y)sin(z) az
View Answer
Answer: a
Explanation: (V = x^2 sin(y)cos(z) )
Gradient of (V = frac{∂V}{∂x} a_x + frac{∂V}{∂y} a_y + frac{∂V}{∂z} a_z )
(= 2x siny cos z a_x + x^2 cos(y)cos(z) a_y – x^2 sin(y)sin(z) a_z )
6. Find the gradient of the function W if W = ρzcos(ϕ) if W is in cylindrical coordinates.
a) zcos(ϕ)aρ – z sin(ϕ) aΦ + ρcos(ϕ) az
b) zcos(ϕ)aρ – sin(ϕ) aΦ + cos(ϕ) az
c) zcos(ϕ)aρ + z sin(ϕ) aΦ + ρcos(ϕ) az
d) zcos(ϕ)aρ + z sin(ϕ) aΦ + cos(ϕ) az
View Answer
Answer: a
Explanation: (W = ρzcos(ϕ) )
Gradient of (W = frac{1}{ρ} frac{∂W}{∂ρ} a_ρ + frac{1}{ρ} frac{∂W}{∂ϕ}a_y + frac{∂W}{∂z}a_z )
(= zcos(ϕ)a_ρ – z sin(ϕ) a_y + ρcos(ϕ) a_z )
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 )
View Answer
Answer: b
Explanation: (A = ρ^2 + z^3 + cos(ϕ) + z )
Gradient of (A = frac{1}{ρ} frac{∂A}{∂ρ} a_ρ + frac{1}{ρ} frac{∂A}{∂ϕ}a_y + frac{∂A}{∂z}a_z )
(=2ρz^3 , a_ρ – frac{1}{ρ} sin(ϕ) , aΦ + 3ρ^2 z^2 + 1 , a_z )
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 )
View Answer
Answer: a
Explanation: (B = ϕln(r)+r^2 ϕ )
Gradient of (B = frac{∂B}{∂r} a_r + frac{1}{r} frac{∂B}{∂θ} a_θ + frac{1}{rsin(θ)} frac{∂B}{∂ϕ} a_Φ )
(= frac{ρ}{r}+ 2rθ ,a_r – r a_θ + frac{lnr}{rsin(θ)} a_Φ )
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 + rfrac{θ}{sin(θ)} a_Φ )
View Answer
Answer: a
Explanation: Gradient of (B = frac{∂B}{∂r} a_r + frac{1}{r} frac{∂B}{∂θ} a_θ + frac{1}{rsin(θ)} frac{∂B}{∂ϕ} a_Φ )
(B = rθϕ)
Hence gradient of (B = θϕ , a_r -ϕ , a_θ + frac{θ}{sin(θ)} a_Φ )
10. If W = x2 y2 + xz, the directional derivative ( frac{dW}{dl} ) in the direction 3 ax + 4 ay + 6 az at (1,2,0).
a) 5
b) 6
c) 7
d) 8
View Answer
Answer: b
Explanation: First find the gradient of W which is (2xy2+z) ax + 2yx2 ay + x az
At (1,2,0) the gradient of W is 8 ax + 4 ay + 1 az
(frac{dW}{dl} = ) (Gradient of W ) . al
( = (8,4,1) . frac{(3,4,6)}{sqrt{(9+16+36)}} )
(=5.88897 = 6.)
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
View Answer
Answer: c
Explanation: The gradient of W is = y ax + (x+z) ay + (y+1) az
At (1,-2,0) the gradient of the function W is -2 ax + ay – az
(frac{dW}{dl} = )(Gradient of W ) . al
(= frac{(-2,1,-1).(3,6,9)}{11.22} )
(= -0.8.)
12. Electric field E can be written as _________
a) -Gradient of V
b) -Laplacian of V
c) Gradient of V
d) Laplacian of V
View Answer
Answer: a
Explanation: Potential difference decreases in the direction of increase in Electric field. Hence Electric field is nothing but the negative of the gradient of potential difference.
13. Let F = (xy2) ax + yx2 ay, F is a not a conservative vector.
a) True
b) False
View Answer
Answer: b
Explanation: Q = xy2 and P = yx2
(frac{∂P}{∂y} = 2xy) and (frac{∂Q}{∂x} = 2xy )
Since, both are equal, F is a conservative vector.
14. State whether the given equation is a conservative vector.
G = (x3y) ax + xy3 ay
a) True
b) False
View Answer
Answer: b
Explanation: P = x3 y and Q = xy3
(frac{∂P}{∂y} = x^3) and (frac{∂Q}{∂x}= y^3 )
Now since they aren’t equal, the vector is not a conservative vector or field.
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 )
View Answer
Answer: a
Explanation: First we have to find the gradient of the function, which is equal to 2x ax + 2y ay + 2z az.
Gradient of f at (2,2,1) is (4,4,2).
(a_n= frac{±(4,4,2)}{6} = frac{2}{3} a_x + frac{2}{3} a_y + frac{1}{3} a_z. )
Global Education & Learning Series – Vector Calculus.
To practice basic questions and answers on all areas of Vector Calculus, here is complete set of 1000+ Multiple Choice Questions and Answers.