1. Gradient of a function is a constant. State True/False.
a) True
b) False
Explanation: Gradient of any scalar function may be defined as a vector. The vector’s magnitude and direction are those of the maximum space rate of change of φ
2. The mathematical perception of the gradient is said to be
a) Tangent
b) Chord
c) Slope
d) Arc
Explanation: The gradient is the rate of change of space of flux in electromagnetics. This is analogous to the slope in mathematics
3. Divergence of gradient of a vector function is equivalent to
a) Laplacian operation
b) Curl operation
c) Double gradient operation
d) Null vector
Explanation: Div (Grad V) = (Del)2V, which is the Laplacian operation. A function is said to be harmonic in nature, when its Laplacian tends to zero.
4. The gradient of xi + yj + zk is
a) 0
b) 1
c) 2
d) 3
Explanation: Grad (xi + yj + zk) = 1 + 1 + 1 = 3. In other words, the gradient of any position vector is 3.
5. Find the gradient of t = x2y+ ez at the point p(1,5,-2)
a) i + 10j + 0.135k
b) 10i + j + 0.135k
c) i + 0.135j + 10k
d) 10i + 0.135j + k
Explanation: Grad(t) = 2xy i + x2 j + ez k. On substituting p(1,5,-2), we get 10i + j + 0.135k.
6. Curl of gradient of a vector is
a) Unity
b) Zero
c) Null vector
d) Depends on the constants of the vector
Explanation: Gradient of any function leads to a vector. Similarly curl of that vector gives another vector, which is always zero for all constants of the vector. A zero value in vector is always termed as null vector(not simply a zero).
7. Find the gradient of the function given by, x2 + y2 + z2 at (1,1,1)
a) i + j + k
b) 2i + 2j + 2k
c) 2xi + 2yj + 2zk
d) 4xi + 2yj + 4zk
Explanation: Grad(x2+y2+z2) = 2xi + 2yj + 2zk. Put x=1, y=1, z=1, the gradient will be 2i + 2j + 2k.
8. The gradient can be replaced by which of the following?
a) Maxwell equation
b) Volume integral
c) Differential equation
d) Surface integral
Explanation: Since gradient is the maximum space rate of change of flux, it can be replaced by differential equations
9. When gradient of a function is zero, the function lies parallel to the x-axis. State True/False.
a) True
b) False
Explanation: Gradient of a function is zero implies slope is zero. When slope is zero, the function will be parallel to x-axis or y value is constant.
10. Find the gradient of the function sin x + cos y.
a) cos x i – sin y j
b) cos x i + sin y j
c) sin x i – cos y j
d) sin x i + cos y j
Explanation: Grad (sin x + cos y) gives partial differentiation of sin x+ cos y with respect to x and partial differentiation of sin x + cos y with respect to y and similarly with respect to z. This gives cos x i – sin y j + 0 k = cos x i – sin y j.