Gradient of a function is a constant. State True/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.
The mathematical perception of the gradient is said to be
Explanation: The gradient is the rate of change of space of flux in electromagnetics. This is analogous to the slope in mathematics.
Divergence of gradient of a vector function is equivalent to
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
Explanation: Grad (xi + yj + zk) = 1 + 1 + 1 = 3. In other words, the gradient of any position vector is 3.
Find the gradient of t = x2y+ ez at the point p(1,5,-2)
Explanation: Grad(t) = 2xy i + x2 j + ez k. On substituting p(1,5,-2), we get 10i + j + 0.135k
Curl of gradient of a vector is
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).
Find the gradient of the function given by, x2 + y2 + z2 at (1,1,1)
Explanation: Grad(x2+y2+z2) = 2xi + 2yj + 2zk. Put x=1, y=1, z=1, the gradient will be 2i + 2j + 2k.
The gradient can be replaced by which of the following?
Explanation: Since gradient is the maximum space rate of change of flux, it can be replaced by differential equations.
When gradient of a function is zero, the function lies parallel to the x-axis. State True/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.
Find the gradient of the function sin x + cos y.
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.