In Calculus, a gradient is a term used for the differential operator, which is applied to the three-dimensional vector-valued function to generate a vector. The symbol used to represent the gradient is ∇ (nabla). For example, if “f” is a function, then the gradient of a function is represented by “∇f”. In this article, let us discuss the definition gradient of a function, directional derivative, properties and solved examples in detail.
Table of Contents:
Gradient Definition
The gradient of a function is defined to be a vector field. Generally, the gradient of a function can be found by applying the vector operator to the scalar function. (∇f (x, y)). This kind of vector field is known as the gradient vector field. Now, let us learn the gradient of a function in the two dimensions and three dimensions.
Gradient of Function in Two Dimensions:
If the function is f(x, y), then the gradient of a function is given by:
Gradient of Function in Three Dimensions:
If the function is f(x, y, z), then the gradient of a function in the three dimensions is given by:
Directional Derivative
The component of the gradient of the function (∇f) in any direction is defined as the rate of change of the function in that direction. For example, the component in “i” direction is the partial derivative of the function with respect to x. In other words, we can say that it is the rate of change of function in the x-direction, by keeping y and z as constant.
Properties of Gradient
The following are the important properties of the gradient of a function:
- The gradient should take a scalar function (i.e., f(x, y) and produces the vector function (∇ f).
- The vector ∇f(x, y) should lie in the plane.
| Also, read: |
Solved Examples on Gradient
Example 1:
Find the gradient of function f(x, y) = x + 3y2.
Solution:
Given function: f(x, y) = x + 3y2
We know that,
= (1+0)i + (0 + 3. 2y2-1)j
= i + 6yj
Hence, the gradient of the function, f(x, y) = x + 3y2 is i + 6yj.
Example 2:
Find the gradient of the function, f(x, y, z) = sin(x)ey ln (z).
Solution:
Given function: f(x, y, z) = sin(x)ey ln (z).
As we know,
Thus,
Therefore, the gradient of the function is:
∇ f(x, y, z) = cos (x) ey ln(z) i + sin(x) ey ln(z)j + sin(x) ey(1/z)k.
Frequently Asked Questions on Gradient
What is meant by the gradient of a function?
The gradient of a function is a vector field. In other words, the gradient is a differential operator applied to the three-dimensional vector valued function to produce a vector field.
Which symbol is used to represent the gradient?
The gradient is represented by the symbol ∇ (nabla).
How to find the gradient of a function?
The gradient of a function can be found by applying the vector operator to the scalar function. I.e., ∇f (x, y).
Comments