How Do You Find a Normal Vector to a Graph?

[RaulTheUCSCSlug]

How do you find a normal vector of a function at a point, such as f(x,y)= ax^y+yx^y^x+b at (X_o,Y_o)

where a and b are just arbitrary constants, and the function is an arbitrary function. So I guess, what is the general steps you take to find the normal? I thought it had to do with the gradient, but I'm still confused.

 

[andrewkirk]

It's a normal to the surface defined by the function, not a normal to the function.

Off the top of my head, one way would be to find the tangent vectors to the surface in the x direction and in the y direction, and take the cross-product..

 

[Ssnow]

You can find the tangent plane of the graph at the point $\left(x_{0},y_{0}\right)$. The coefficients of the tangent plane $n_{1}x+n_{2}y+n_{3}z+c=0$ are the component of the normal vector $\vec{n}=\left(n_{1},n_{2},n_{3}\right)$ of the plane that is a normal vector of $f(x,y)$ at the given point ...

 

[HallsofIvy]

Given a surface described by z= f(x, y),the surface can be thought of as a "level surface for the function $\phi(x, y, z)= f(x, y)- z= 0$ so $\nabla (f(x,y- z)= f_x\vec{i}+ f_y\vec{j}- \vec{k}$ is immediately a vector normal to the surface.