Question

What is the purpose of gradient tool? State the use of four types of gradient learned in this chapter.

Answer 1

Gradient is the multidimensional rate of change of given function.

"Gradient vector is a representative of such vectors which give the value of differentiation (means characteristic of curve in terms of increasing & decreasing value in 3 or multi dimensions) in all the 360° direction for the given point on the curve"

We know that the vector representation is in form of unit vector of x,y,z. So that a vector is always made of x,y,z components So same method can be applied for the gradient But in this case the x,y,z components are little bit different First take the projection of given 3 dimensional curve [z= f(x,y)] into plane x,z so that it means constant y .

Now take differentiation of a=f'(x) at constant y . So this 'a' is 'x' component of gradient vector.(so partial differentiation is nothing but differentiating on projected plane of curve).By following this method y,z both can be got
Here is video for visualization for the above
If we take a dot product between gradient and vector we can get the increasing or decreasing characteristic of curve in x direction by dot product.

So if we want to get the increasing or decreasing characteristic in direction (x,y,z) by its dot product with gradient
v ector. So based on this we can say that we have converted the whole system characteristic from scalar form to the vector form.

See also : Vector Calculus: Understanding the Gradient