Question

My second time on this site. Please help. Does anyone know if this is a typo. Its OCW Scholar Multivariable calculus Session 62. The PDF title is "Gradient Fields". Page 2 midpage!. F = del ln times square root of x^2 + y^2. Shouldn't that be 1/2 of del ln times square root of x^2 + Y^2? When I differentiate with respect to x and then y there is a 2 in the denominator so don't you need to take it out with the 1/2? Or am I really missing the boat on this? Thank you guys and gals. Julius

Answer 1

I think there is a typo (missing minus sign) where they take the partials (but it does not matter much). Specifically
\[ \frac{\partial}{\partial x } \left( \frac{y}{x^2+y^2}\right) \\
=\frac{\partial}{\partial x } y(x^2+y^2)^{-1} = -y(x^2+y^2)^{-2} 2 x\\
=\frac{-2xy}{(x^2+y^2)^2}
\]
similarly for the other component (there is a missing minus sign)

Answer 2

as for the gradient of \( \ln\sqrt{x^2+y^2} \)
we use
\[ d \ln u = \frac{1}{u} \ du\]
the first component is
\[ \frac{\partial}{\partial x} (x^2+y^2)^\frac{1}{2} \\
= \frac{1}{(x^2+y^2)^{\frac{1}{2}}} \cdot \frac{1}{2} (x^2+y^2)^ {-\frac{1}{2}}\cdot 2x\\
= \frac{x}{x^2+y^2}
\]
similarly for the other component

Answer 3

Thank you so much!!! you are so right and so kind! off to the next section! Have a good day