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