line integral along a curve Let $ \varphi(x,y) = x^3y+xy^3 ((x,y) \in R^")$, and let $C$ the curve given by $\varphi(x,y)=5$
The question is, how can I calculate the line integral of $\bigtriangledown\varphi$ along the curve $C$?
 A: Well you are on a curve of constant value, so the gradient's component along this line iz zero.
A: $\newcommand{\+}{^{\dagger}}
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\begin{align}
\color{#00f}{\large\int\pars{\partiald{\varphi}{x}\,\dd x + \partiald{\varphi}{y}\,\dd y}}
&=
\int\pars{\partiald{\varphi}{x} + \partiald{\varphi}{y}\,y'}\dd x
=\int\bracks{\partiald{\varphi}{x} + \partiald{\varphi}{y}\,
\pars{-\,{\partial\varphi/\partial x \over \partial\varphi/\partial y}}}\dd x
=\color{#00f}{\Large 0}
\end{align}
