- #1
daftjaxx1
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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?