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The1TL
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Homework Statement
Suppose f:R^2 - {0} → R is a differentiable function whose gradient is nowhere 0 and that satisfies -y(df/dx) + x(df/dy) = 0 everywhere.
a) find the level curves of f
b) Show that there is a differentiable function F defined on the set of positive real numbers so that f(x) = F(||x||)
Homework Equations
The Attempt at a Solution
a) I know that gradient vectors are orthogonal to level curves. So the fact that -y(df/dx) + x(df/dy) = 0 seems to show that the gradient vector is orthogonal to any vector of the form (x, -y). So would all vectors of this form be the level curves?
b)could I just show that F is a function that is the same as f but multiplies results that would be negative by -1?