Gradient of a scalar field, Hello friends, today I will talk about the gradient of a scalar field.

Have a look!!

### Gradient of a scalar field

The gradient of any scalar field is always used in a short form called ‘grad’.

Suppose is a scalar field. Then the gradient of the scalar field is ‘grad ’. It also has a mathematical notation. That is, .

The standard formula for the gradient of any scalar field is

Now we need to know about it because we have to use it several times in vector analysis.

One of the immediate uses will be in the directional derivative of any scalar function.

Just have a look at these two examples!!

#### Examples of the gradient of a scalar field

Note: None of these examples are mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.

So here is the first example.

##### Example 1

According to Stroud and Booth (2011)* “If find, at the point the values of (a) grad and (b) |grad |.”

##### Solution

Here the given scalar field is I will start with part (a) of this example.

###### (a)

To get the value of grad , I will start with the partial differentiation.

###### Step 1

First of all, I’ll differentiate partially with respect to . Now for that, I’ll use the same technique as the first-order partial derivative of functions with three variables.

So will be

Now this means

So it will be

Thus I can say that

Now I’ll get

So it will be

Now this means

So will be

Thus I can say that

Similarly, I’ll get the value of . Thus will be

So it means

Thus it will be

Hence I can say

Next, I’ll get the values of at the point .

###### Step 2

So at the point the values of will be

and

Thus the gradient of the scalar field will be

Hence I can conclude that is the answer to part (a) of this problem.

Now I will go to part (b).

###### (b)

Here I need to get the value of |grad |.

Now |grad | means the magnitude of the gradient vector grad .

Therefore at the point the value of |grad | will be

Now I’ll simplify it.

So it will be

Now I already know that .

Thus it will be

Hence I can conclude that this is the answer to part (b) of this problem.

Now I move to the next example.

##### Example 2

According to Stroud and Booth (2011)* “Determine grad where and obtain its value at the point ”

##### Solution

Now here the given scalar field is

First of all, I have to determine the value of grad . Next, I need to find its value at the point

Therefore following my earlier example, here also I will start with the partial differentiation.

###### Step 1

So will be

So this means

Now I’ll get Thus it will be

So this means

Now this gives

Similarly, will be

So this means

which gives

Therefore I can say that

Thus the gradient of the scalar field is

Now I will do the next step.

###### Step 2

Here I will find the value of grad at the point

Let’s say that at the point the value of grad will be .

So will be

Now I’ll simplify it to get

which means

But I’ll still simplify it a bit more to get

Hence I can conclude that the gradient of the scalar field is

Also its value at the point is

These are the answers to the second example.

So here ends my second as well as the last example.

Dear friends, this is the end of my today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

###### *Reference: K. A. Stroud and Dexter J. Booth (2011): Advanced engineering mathematics, Industrial Press, Inc.; 5th Edition (March 8, 2011), Chapter: Vector analysis 1, Further problems 22, p. 815, Q. No. 15 (Example 1); Q. No. 21 (Example 2).

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