Directional Derivative of w on Intersection of Surfaces

[alvielwj]

Homework Statement

Find the directional derivative of w = x^3 + y^2 + 3z at the point (1, 3, 2) on the curve of intersection of the surfaces
x^2 + y^2− 2xz = 6, 3x^2− y^2 + 3z = 0, in the direction of increasing z.

Homework Equations

The Attempt at a Solution

i know how to solve the directional dirivative of a function of at a point on the given vector.but in this question I don't know how to solve the equations,and "in the direction of increasing z" does not make sense

 

[Dick]

The tangent vector along the curve is perpendicular to the gradient vectors of the two surfaces. Find them at (1,3,2). Now that you have these two vectors, how do you find one that is perpendicular to them both?

 

[alvielwj]

cross product of these two vectors

 

[alvielwj]

But I can't get the meaning of "in the direction of increasing z"

 

[Dick]

But I can't get the meaning of "in the direction of increasing z"

Using your cross product you'll get a vector t for the tangent vector. But -t is also a tangent. Pick the one that has positive z component.

 

[alvielwj]

I want to get the gradient vector of x^2 + y^2− 2xz = 6,so z=(x^2+y^2-6)/2x
,is the gradient vector (dz/dx dz/dy) but i should have three dimensions?

 

[Dick]

If f=x^2+y^2-2xz-6=0 is your surface the gradient vector is (df/dx,df/dy,df/dz) (all partial derivatives). Of course you get three components.

 

[alvielwj]

Why do we calculate the gradient vectors of the surfaces? What is the geometrical meaning of gradient vector? I think I should calculate the normals of two surfaces and get the cross product of two normals, because the vector of the intersection curve is perpendicular to both of the normals..

 

[Dick]

The normal vector points in the same direction as the gradient vector. Don't you calculate the normal vector using the gradient vector??

 

[alvielwj]

AH, finally,i figured it out , thank a lot