Question

Can anyone explain what r stands for in P-Set 8, problems 4A-1, 2, and 3? For instance, problem 4-A-2(b) is:
- find the gradient field of w=ln(r).
Is r the position vector ? In that case, wouldn't the function w really be a vector function with component ln(x) and ln(y)? Yet, the answer given in the solutions is that the gradient field of w = (xi+yj)/r^2

Answer 1

r is the distance from the origin (the "r" component of polar coordinates ( r, \(\theta\) )
\[ r= \sqrt{x^2+y^2}\]
\[ w= \ln r \\ \nabla w=\left< \frac{\partial}{\partial x} \ln r, \frac{\partial}{\partial y} \ln r\right>\\
\nabla w= \left<\frac{1}{r} \frac{\partial }{\partial x} \ln (x^2+y^2)^{\frac{1}{2}},\frac{1}{r} \frac{\partial }{\partial y} \ln (x^2+y^2)^{\frac{1}{2}}\right>\]
finish taking the partial derivative with respect to x (and y) to get the result

Answer 2

Thanks. I had thought of that, but made a silly mistake on differentiating and got the wrong result. Problem solved!