Unit Normal to a level surface

In summary: That explains why I couldn't seem to get it to work for some scalars. Thanks for your help!In summary, the goal is to calculate the unit normal to the level surface φ = 0 at the point r = (0,1,0). Using the formula for the unit normal, it is found that the normal vector is equal to the y unit vector. However, when trying to use the partial derivatives method, there is an indeterminate value for the partial derivative w.r.t y. This is because the surface has a vertical tangent plane at the point, but the unit normal still exists. Therefore, it is best to use the formula for the unit normal instead of the partial derivatives method in this case.
  • #1
CRD_98
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Homework Statement



Given $$\phi = x^{2} +y^{2}-z^{2}-1 $$

Calculate the unit normal to level surface φ = 0 at the point r = (0,1,0)

Homework Equations


  1. $$ \hat{\mathbf n} = \frac{∇\phi}{|\phi|}$$
  2. $$ z = \sqrt{x^{2}+y^{2} -1} $$
  3. $$ \mathbf n = (1,0,(\frac{\partial z}{\partial x})_{P}) \times (0,1,(\frac{\partial z}{\partial y})_{P}) $$

The Attempt at a Solution



Using equation (1) above I obtained:
$$\hat{\mathbf n} = \frac{1}{\sqrt{x^{2} + y^{2} +z^{2}}} (x\hat{\mathbf i}, y\hat{\mathbf j}, -z\hat{\mathbf k})$$

Evaluating this at (0,1,0) resulted in $$\hat{\mathbf n} = \hat{\mathbf j}$$

As far I can tell this is correct? I ran into a problem when I tried using equations (2) and (3) to check my answer;
I worked out the partial derivatives as follows:

$$\frac{\partial z}{\partial x} = \frac{x}{\sqrt{x^{2}+y^{2}-1}} $$
$$\frac{\partial z}{\partial y} = \frac{y}{\sqrt{x^{2}+y^{2}-1}} $$

Evaluating these partial derivatives at (0,1,0) gave:

$$(\frac{\partial z}{\partial x})_{P} = 0 $$

$$(\frac{\partial z}{\partial y})_{P} = \frac{1}{\sqrt{1^{2}-1}} = \frac{1}{0} $$

My problem is that I am getting an indeterminate value for the partial derivative w.r.t y evaluated at (0,1,0). I'm struggling to see where I have gone wrong, if anyone could help point out my mistake it would be greatly appreciated :)

I have also attached a wolfram alpha screenshot in relation to my second attempt's method, does this second method only work for some scalars?
 

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  • #2
It's best to use the formula ##\nabla \phi = \langle \phi_x,\phi_y, \phi_z \rangle##. Your answer is correct. The problem is probably that the surface has a vertical tangent plane at the point whose normal is in the y direction. z isn't a function of x and y at that point, but the surface still has a unit normal.
 
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  • #3
LCKurtz said:
It's best to use the formula ##\nabla \phi = \langle \phi_x,\phi_y, \phi_z \rangle##. Your answer is correct. The problem is probably that the surface has a vertical tangent plane at the point whose normal is in the y direction. z isn't a function of x and y at that point, but the surface still has a unit normal.

Ah okay thank you for that, I'll stick to the grad method
 

Related to Unit Normal to a level surface

1. What is the unit normal to a level surface?

The unit normal to a level surface is a vector that is perpendicular to the surface at a given point. It represents the direction in which the surface is changing the most rapidly.

2. How is the unit normal calculated?

The unit normal is calculated by taking the gradient of the function that defines the level surface at a specific point. The gradient is then normalized to have a magnitude of 1, representing the direction of the unit normal vector.

3. What is the significance of the unit normal to a level surface?

The unit normal is important in understanding the geometry and behavior of the level surface. It can be used to determine the rate of change or slope of the surface at a given point, and is also used in many applications in physics and engineering.

4. Can the unit normal be negative?

Yes, the unit normal can have a negative direction, as it represents a vector and can point in any direction. However, its magnitude will always be 1, as it is a unit vector.

5. How does the unit normal relate to the curvature of a level surface?

The unit normal is directly related to the curvature of a level surface. The curvature is determined by the rate of change of the unit normal vector, and the direction of the unit normal vector determines the principal direction of curvature.

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