- #1

flyingpig

- 2,579

- 1

## Homework Statement

Let's say I have a function for a circle

[tex]x^2 + y^2 = C[/tex] where C is a constant.

Then this is a cylinder with the z-axis.

Now in my ODE book, we would normally define it as

[tex]F(x,y) = C = x^2 + y^2[/tex] as a level surface.

Now my question is about what the partial derivative with respect to x mean as opposed to (single-variable calculus) derivative with respect to x mean. Am I losing anything if I take one derivative over the other?

I should mention that many of these problems assume that F(x,y(x)).

[tex]\frac{\partial F}{\partial x} = 2x[/tex]

[tex]\frac{\partial F}{\partial y} = 2y[/tex]

[tex]\frac{\mathrm{d} F}{\mathrm{d} x} = 2x + 2y\frac{\mathrm{d} y}{\mathrm{d} x} = 0[/tex]

So now my question is, what exactly is this

as opposed to[tex] 2x + 2y\frac{\mathrm{d} y}{\mathrm{d} x} = 0[/tex]

2x