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## Main Question or Discussion Point

I'm trying to solve this problem:

Compute [tex]\oint_c(y+z)dx + (z-x)dy + (x-y)dz [/tex] using Stoke's theorem, where c is the ellipse [tex]x(t) = asin^2t, \ y(t) = 2asintcost, z(t) = acos^2t, 0\leq t \leq \pi [/tex]

The version of stoke's theorem I learned is:

[tex]

\int_c \overrightarrow{F} \cdot d\overrightarrow{r}

= \int_s curl \overrightarrow{F} \cdot d\overrightarrow{S}

=\iint_s curl \overrightarrow{F}\cdot \overrightarrow{n} \cdot dS

[/tex]

where S is the elliptical surface bounded by the curve c, F is a vector field and n is the unit vector pointing out at that point.

In this case, [tex]F = <y+z, z-x, x-y>[/tex], and I calculated curl F to be [tex]<-2, 0, -2>[/tex].

So we have to find

[tex]\iint_s <-2, 0, -2> \cdot \overrightarrow{n} \cdot dS [/tex]

How would I find [tex]\overrightarrow {n} [/tex] and dS, and also the bounds of integration for the double integral?

Compute [tex]\oint_c(y+z)dx + (z-x)dy + (x-y)dz [/tex] using Stoke's theorem, where c is the ellipse [tex]x(t) = asin^2t, \ y(t) = 2asintcost, z(t) = acos^2t, 0\leq t \leq \pi [/tex]

The version of stoke's theorem I learned is:

[tex]

\int_c \overrightarrow{F} \cdot d\overrightarrow{r}

= \int_s curl \overrightarrow{F} \cdot d\overrightarrow{S}

=\iint_s curl \overrightarrow{F}\cdot \overrightarrow{n} \cdot dS

[/tex]

where S is the elliptical surface bounded by the curve c, F is a vector field and n is the unit vector pointing out at that point.

In this case, [tex]F = <y+z, z-x, x-y>[/tex], and I calculated curl F to be [tex]<-2, 0, -2>[/tex].

So we have to find

[tex]\iint_s <-2, 0, -2> \cdot \overrightarrow{n} \cdot dS [/tex]

How would I find [tex]\overrightarrow {n} [/tex] and dS, and also the bounds of integration for the double integral?