# Steepest tangent vector to an explicit surface

A steepest tangent vector to the explicit surface $z=f(x,y)$ at the fixed point $(x,y,f(x,y))$ is given by the formula: $$\mbox{steepTan} = (?)(?)(?)$$ You can replace the $?$s with formulas like $$\frac{\partial}{\partial x}[f(x, y)]$$ One of the hints is that the steepest tan is the $(dx,dy,?)$. The hardest part is figuring out the third variable. Which can be figured out by $$dz = mdx +ndy$$

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What exactly is the question? – Matt Mar 11 '11 at 2:13
I guess the question would be what are the 3 parts of the steepest tangent vector to an explicit surface. I understand that it would be the gradient but what would the 3rd component be? (f/x[x,y],f/y[x,y],______) – user8093 Mar 11 '11 at 2:28

$$\left(x + \lambda\frac{\partial f}{\partial x},y + \lambda\frac{\partial f}{\partial y},z\left(x + \lambda\frac{\partial f}{\partial x},y + \lambda\frac{\partial f}{\partial y}\right)\right)\;,$$
so the tangent vector, the derivative with respect to $\lambda$, is
$$\left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial x}\cdot\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y}\cdot \frac{\partial f}{\partial y}\right)=\left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\lvert\nabla f\rvert^2\right)\;.$$