I would like a hint on how to begin a vector calculus problem

[Michael King]

Homework Statement

I am just confused about how the question is structured, and I am unsure on how to get the relevant information to answer the question:

Find the angle between the surface normal directions of $$r^{2} = 9$$ and $$x + y + z^{2} = 1$$ at the joined point (2,-2,1)

Homework Equations

I am thinking both the Scalar and Vector product rules, possibly in component form.

The Attempt at a Solution

I begin by stating the definition of $$r^{2}$$:

$$r^{2} = x^{2} + y^{2} + z^{2}$$

I draw a sphere on paper and indicate the direction of the radius from the origin to the joined point and put the r unit vector following the direction of the radius, and I am really stuck on how to express $$x + y + z^{2} = 1$$, and to bring the two together. It has been a LONG summer, and I was thinking that the x,y,z terms could be the components but isn't that function a scalar function? How would I get the normal direction for that? I was thinking of taking the grad of the function then taking the scalar product of the two, but I am stuck on expressing it.

 

[lanedance]

So the question has two parts
- find the normal vectors to each of the surfaces given at the point p = (2,-2,1)
- what is the angle between the two vectors

For the first part:

as you hinted r^2 = 9 is an equation of a sphere with centre 0 & radius 3.

The normal to the surface, at any point p on the surface, will be parallel to the radius vector from centre 0 = (0,0,0) to p

As for the other surface you can write the surface as the level curve of F(x,y,z)
F(x,y,z) = x + y + z^2 - 1 = 0

NOw consider a tangent plane to the surface at a point p on the surface.

As F(x,y,z) is a level curve, the rate of change of F for movement in any direction in the tangent plane will be zero (why?).

Any ideas what you can use to find the direction of maximium rate of change of F, and how this relates to the last comment?