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3 votes

Evaluate the line integral of $\textbf{F}(x,y,z) = (f(x),g(y),h(z))$ along the curve $x^{2/3} + y^{2/3} = 1$

Based on the proposed conditions, it results that \begin{align*} \langle\textbf{F}(\textbf{r}(t)),\textbf{r}'(t)\rangle & = -3\sin(t)\cos^{2}(t)f(\cos^{3}(t)) + 3\sin^{2}(t)\cos(t)g(\sin^{3}(t)) \...
Átila Correia's user avatar
2 votes

Line integral of a vector field over a triangle

The $\text{s}$ in $d\text{s}$ stands for surface. It's a surface integral as stated by @Sebastiano . But if you wanted to compute the line integral over the triangle's border (with $d\text{r}$), your ...
Tetrahydron's user avatar
1 vote

Do line integrals of closed curves depend on the orientation of the curve or of the vector field?

The double integral in Green's Theorem is only equal to the line integral over the positively oriented boundary. That is the statement of the theorem. The clockwise orientation is not the positively ...
whpowell96's user avatar
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