Test: Gradient

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.

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

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.

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

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

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).

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

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

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.

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.