Let φ(x, y) = arctan (y/x) .
(a) find the vector field f : = ∇φ, and simplify its components as much as possible.
(b) what is the domain of definition of f?
(c) find all the points (a, b) (belonging to the domain of definition of f) at which the vector f(a, b) is parallel to the line y = x. (note: for a real number u, arctan u is also denoted by tan−1 u. use the notation that you prefer.)

a)

b)

c) the points of the form (x, -x) for x≠0

Step-by-step explanation:

a)

If φ(x, y) = arctan (y/x), the vector field F = ∇φ would be

On one hand we have,

On the other hand,

So

b)

The domain of definition of F is  

i.e., all the plane X-Y except the (0,0)

c)

Here we want to find all the points such that

where k is a real number other than 0.

But this means

So, all the points in the line y = -x except (0,0) are parallel to the vector field F, that is, the points (x, -x) with x≠ 0

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