Math Problem

[Narmer]

Let f(x, y) = x^2 + y^2 and g(x, y) = x^3 + y^3. For every point (x, y), let
d(x, y) be the dimension of the linear span of the gradient vectors of f, and
of g, at (x, y). Compute d(x, y) for all (x, y).

The problem I'm having here is that the gradient vectors for f and g occupy R^2 but f is a first order equation and g is second order. How do I set this problem up?

Do your own homework!

AnandTech Moderator

 

[Narmer]

^

 

[IronWing]

Let me state that I am no longer in school so that you'll understand my reply.

HAHAHAHAHAHAHAHAHAHA!!!!!!!!!!!!!

 

[Narmer]

Originally posted by: ironwing
Let me state that I am no longer in school so that you'll understand my reply.

HAHAHAHAHAHAHAHAHAHA!!!!!!!!!!!!!

Thanks for nothing🙂

But, more seriously, I'm going to guess that the basis here is [1 -1]^T.

 

[dogooder]

What is the gradient of f at (x,y)? What is the gradient of g at (x,y)? What is the dimension of the span of these two vectors? Seems straightforward to me.

 

[Narmer]

Originally posted by: dogooder
What is the gradient of f at (x,y)? What is the gradient of g at (x,y)? What is the dimension of the span of these two vectors? Seems straightforward to me.

I all does seem straightforward, that's why I posted what I believed is the answer in my last post. Am I correct? If I am not, then please explain the answer.

 

[dogooder]

I don't understand your answer. How does it answer the question?

Anyway... The answer should depend on (x,y), shouldn't it? That is, you figure out the two gradient vectors, then see when their span has dimension 0. Then, see when their span has dimension 1. Then, the remainder of the time their span must have dimension 2.

 

[Narmer]

Originally posted by: dogooder
I don't understand your answer. How does it answer the question?

Anyway... The answer should depend on (x,y), shouldn't it? That is, you figure out the two gradient vectors, then see when their span has dimension 0. Then, see when their span has dimension 1. Then, the remainder of the time their span must have dimension 2.

the gradient vector for f is 2xi +2yj. For g, it is 3x^2i + 3y^2j. I believe both gradient vectors would span the nullspace iff the basis was the zero vector or [1 -1]. I don't think I understand your last two sentences.