The Gradient and Directional Derivative

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Introduction:

Gradient: In vector calculus, the gradient is considered as vector field in a function.It points to the points in the route of the maximum rate of increase of the scalar field. Its magnitude is the maximum rate of modify.

Directional derivative: Directional derivative represents the instantaneous rate of modification of the function. It generalizes the view of a partial derivative.

Gradient:

The gradient is defined for the function f(x,y) is as

gradf(x,y)= [gradf(x,y)] = [(delf)/(delx)] i + [(delf)/(dely)] j

This can be calculated by putting the vector operator r to the f(x,y) which is scalar function. That vector field is called as gradient vector field.

Example:

Find the gradient of the function f(x,y)=x+y.

Solution:

Given function is f(x,y)=x+y

[gradf(x,y)] = [(delf)/(delx)i] + [(delf)/(dely)j]

= [(del)/(delx)] (x+y)i+ [(del)/(dely)] (x+y)j

=(1+0)i+(0+1)j

= i+j

Therefore the gradient of the function f(x,y)=x+y is i+j.

Directio...

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