- #1

Tsunoyukami

- 215

- 11

*If ∇ x*(from Walter Strauss'

**v**=**0**in all of three dimensional space, show that there exists a scalar function ##\phi (x,y,z)## such that**v**= ∇##\phi##.*Partial Differential Equations*, 2nd edition; problem 11; pg 20.)

I'm not really sure where to begin with this problem. I asked a few of my friends in math and they all provided me with suggestions that are above my level (ie. using something called Poincare's Lemma which we haven't covered in any of my classes yet). My difficulty lies in proving this is true in the direction required.

I can easily show this is true in the reverse direction. If ∇##\phi## =

**v**, then ∇ x

**v**= 0. I can show this with relative ease by simply writing out each component explicitly. (I won't do this here, however, since that's not what the question is asking me to do). I'm not sure how to go in the reverse direction, however. Any help would be greatly appreciated

(Also, to put the problem into context: this problem is an assigned problem near the beginning of an introductory PDEs class that assumes a familiarity with vector calculus, basic linear algebra and ODEs.)

It also might be worth mentioning that I think we can assume ##\phi## is a "nice" function - that is, that it is continuous and twice differentiable. Thanks a bunch in advance for any nudges in the right direction.