Normal Vector for (x,y,z) Surface of f(x,y)

[EV33]

Homework Statement

My question is if this is the formula for a normal vector for the point (x,y,z) of a surface of some function f(x,y).

(x,y,z)+t(f$$_{x}$$,f$$_{y}$$,-1)


My teacher used it in class and I just wanted to know if it is what I think it is.

Homework Equations

The Attempt at a Solution

Thank you.

 

[EV33]

Those superscripts should be subscripts.

 

[arkajad]

Your surface is (x,y,f(x,y)). Take a path in (x,y) plane: (x(s),y(s)) - s is the parameter along the path. The path on your surface is (x(s),y(s),f(x(s),y(s))). Differentiate at s=0, denote the derivative by a dot. You get for the tangent vector:

$$({\dot x},{\dot y},{f_x{\dot x}+f_y{\dot y})$$

Take the scalar product with $$(f_x,f_y,-1)$$ - you will get automatically 0. So you get a vector that is orthogonal to the surface. You may like to normalize its length - multiply by an appropriate number t, and then move it to the point on the surface (x,y,z=f(x,y)). The result is as in your formula.

 

[HallsofIvy]

If z= f(x,y), then we can think of the surface as a "level surface" for some function F(x, y, z)= f(x,y)- z. The gradient $\nabla F= f_x\vec{i}+ f_y\vec{j}- \vec{k}$ is always normal to a level surface and so is a normal vector for that surface.

What you have is the vector equation for a line perpendicular to the surface. For fixed t it gives a normal vector.