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

**The 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

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

Let’s say it as

And finally the unit vector normal to the surface

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

**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

**Step 1**

Now, to get the value of grad

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

First of all, I’ll differentiate

So

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

Thus

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

Thus

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

which means

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

**Step 4**

So at the point,

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

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

**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

First of all, I’ll differentiate

So

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

Thus

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

Thus

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

which means

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

**Step 4**

So at the point

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

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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