Test: Gradient

QUESTION: 1

Gradient of a function is a constant. State True/False.

A.
True
B.
False

Solution:

Answer: b
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.

QUESTION: 2

The mathematical perception of the gradient is said to be

A.
Tangent
B.
Chord
C.
Slope
D.
Arc

Solution:

Answer: c
Explanation: The gradient is the rate of change of space of flux in electromagnetics. This is analogous to the slope in mathematics.

QUESTION: 3

Divergence of gradient of a vector function is equivalent to

A.
Laplacian operation
B.
Curl operation
C.
Double gradient operation
D.
Null vector

Solution:

Answer: a
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.

QUESTION: 4

4. The gradient of xi + yj + zk is

A.
0
B.
1
C.
2
D.
3

Solution:

Answer: d
Explanation: Grad (xi + yj + zk) = 1 + 1 + 1 = 3. In other words, the gradient of any position vector is 3.

QUESTION: 5

Find the gradient of t = x²y+ e^z 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

Solution:

Answer: b
Explanation: Grad(t) = 2xy i + x² j + e^z k. On substituting p(1,5,-2), we get 10i + j + 0.135k

QUESTION: 6

Curl of gradient of a vector is

A.
Unity
B.
Null vector
C.
Zero
D.
Depends on the constants of the vector

Solution:

Answer: c
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).

QUESTION: 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

Solution:

Answer: b
Explanation: Grad(x²+y²+z²) = 2xi + 2yj + 2zk. Put x=1, y=1, z=1, the gradient will be 2i + 2j + 2k.

QUESTION: 8

The gradient can be replaced by which of the following?

A.
Maxwell equation
B.
Volume integral
C.
Differential equation
D.
Surface integral

Solution:

Answer: c
Explanation: Since gradient is the maximum space rate of change of flux, it can be replaced by differential equations.

QUESTION: 9

When gradient of a function is zero, the function lies parallel to the x-axis. State True/False.

A.
True
B.
False

Solution:

Answer: a
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.

QUESTION: 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

Solution:

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.