Unit normal vector to any surface. Hello friends, today it’s about the unit normal vector to any surface. Have a look!!

#### Unit normal vector to any surface

Let’s suppose there’s a surface . And I have to find out a unit normal vector to this surface at the point, say, .

Now the question is how to do it??

So, first of all, I’ll find out the gradient of the surface. Let’s say the gradient is . Then I have to find out the value of at the point .

Let’s say it as . Then I’ll determine the magnitude of the vector , that is .

And finally the unit vector normal to the surface at the point will be

So now I’ll give some examples on that.

#### Solved examples of the unit normal vector to any surface

Note: None of these examples is mine. I have chosen these from some 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)* “Find the unit normal to the surface at the point .”

##### Solution

Let’s say, the surface is

First of all, I’ll find out the gradient of this surface. So this means I’ll get the value of grad of this scalar field.

###### Step 1

Now, to get the value of grad , I will start with the partial differentiation of the surface .

You can also check out: First-order partial derivative of functions with three variables

First of all, I’ll differentiate partially with respect to to get .

So will be

Thus it means

Now I’ll use the same technique as the first-order partial derivative of functions with two variables.

Therefore it will be

Hence I can say

Next, I’ll get the value of

###### Step 2

So now I’ll differentiate partially with respect to to get .

Thus will be

Hence it means

Now I’ll use the same technique as in Step 1.

Therefore it will be

Hence I can say

Next, I’ll get the value of

###### Step 3

So now I’ll differentiate partially with respect to to get .

Thus will be

Hence it means

Now I’ll use the same technique as in Step 1.

Therefore it will be

Hence I can say

Thus I can say that the gradient of the surface is

which means

Now I’ll get the unit normal vector of this surface at the point .

###### Step 4

So at the point , the gradient of the surface will be

Now I’ll simplify it to get

which gives

Next, I’ll get the magnitude of the gradient vector .

Thus it will be

Then I’ll simplify it to get

which gives

So, as per the formula, I can say that the unit normal to the given surface at the point is

Now this means

Hence I can conclude that this is the answer to the first example.

Now I’ll give another example.

##### Example 2

According to Stroud and Booth (2011)* “Determine the unit vector normal to the surface at the point .”

##### Solution

Let’s say, the surface is

Now here also I’ll find out the gradient of this surface.

###### Step 1

In order to get the value of grad , I will start with the partial differentiation of the surface .

First of all, I’ll differentiate partially with respect to to get .

So will be

Thus it means

Now I’ll use the same technique as the first-order partial derivative of functions with two variables.

Therefore it will be

Hence I can say

Next, I’ll get the value of

###### Step 2

So now I’ll differentiate partially with respect to to get .

Thus will be

Hence it means

Now I’ll use the same technique as in Step 1.

Therefore it will be

Hence I can say

Next, I’ll get the value of

###### Step 3

So now I’ll differentiate partially with respect to to get .

Thus will be

Hence it means

Now I’ll use the same technique as in Step 1.

Therefore it will be

Hence I can say

Thus I can say that the gradient of the surface is

which means

Now I’ll get the unit normal vector of this surface at the point .

###### Step 4

So at the point , the gradient of the surface will be

Now I’ll simplify it to get

which gives

Next, I’ll get the magnitude of the gradient vector .

Thus it will be

Then I’ll simplify it to get

which gives

So, as per the formula, I can say that the unit normal to the given surface at the point is

Now this means

Hence I can conclude that this is the answer to the first 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!!

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