Question regarding unit normal of some function

[keensaj]

Please can you help me with this question.

A surface h in the three dimensions is given by the equation z=g(x,y)

Find the unit normal at any point h in two separate ways:

a)By identifying h with an equipotential for some scalar field φ to be determined and computing ∇φ.

b)By expressing the points of h in parametric form r(x, y) and computing ∂r/dx × ∂r/dy

I think that you have to write h(x,y,z)=xi+yj+g(x,y)k
but can someone explain what a) means
thanks

 

[SteamKing]

Please can you help me with this question.

A surface h in the three dimensions is given by the equation z=g(x,y)

Find the unit normal at any point h in two separate ways:

a)By identifying h with an equipotential for some scalar field φ to be determined and computing ∇φ.

b)By expressing the points of h in parametric form r(x, y) and computing ∂r/dx × ∂r/dy

I think that you have to write h(x,y,z)=xi+yj+g(x,y)k
but can someone explain what a) means
thanks

This article explains the mechanics behind calculating ∇φ and how that can be used to find the unit normal to a surface:

http://tutorial.math.lamar.edu/Classes/CalcIII/DirectionalDeriv.aspx

BTW, ∇φ is called the 'gradient' of the function φ.

The symbol ∇ is known as an 'operator' and means ∇ = ∂ / ∂x + ∂ / ∂y + ∂ / ∂z

https://en.wikipedia.org/wiki/Gradient

Much of this won't make sense if you haven't studied some elementary vector calculus.

 

[HallsofIvy]

If z= g(x,y), then $\phi(x,y,z)= z- g(x,y)= 0$, a constant. That is the "equipotential" referred to in (a). Since g is a constant function, it gradient, $\nabla \phi= \frac{\partial \phi}{\partial x}\vec{i}+ \frac{\partial \phi}{\partial y}\vec{j}+ \frac{\partial \phi}{\partial z}\vec{k}$, is perpendicular to surface at every point.