# Thread: Vector Calculus

1. ## Vector Calculus

Show that if $\displaystyle \sigma$(x,y) satisfies Laplace's equation,

$\displaystyle \frac{\partial \sigma}{\partial x^2}+\frac {\partial \sigma}{\partial y^2}=0$ on a simply connected region R, then $\displaystyle \forall$closed curves C of R, we have
$\displaystyle \int_c (\sigma_y dx - \sigma_x dy) =0$

2. Originally Posted by Andreamet
Show that if f(x,y) satisfies Laplace's equation,

$\displaystyle \frac{\partial \sigma}{\partial x^2}+\frac {\partial \sigma}{\partial y^2}=0$ on a simply connected region R, then $\displaystyle \forall$closed curves C of R, we have
$\displaystyle \int_c (\sigma_y dx - \sigma_x dy) =0$
Use greens theorem

$\displaystyle \iint_D -\sigma_{xx}-\sigma_{yy}dA=-\iint_D 0 dA=0$

3. Originally Posted by TheEmptySet
Use greens theorem

$\displaystyle \iint_D -\sigma_{xx}-\sigma_{yy}dA=-\iint_D 0 dA=0$

Hi, thanks a lot. However, from the Question, I know that we need to apply Green's theorem. However, I am confused about what you did. I am very confused

4. Originally Posted by Andreamet
Hi, thanks a lot. However, from the Question, I know that we need to apply Green's theorem. However, I am confused about what you did. I am very confused

The formal statement of Green's theorem is (Stuart Calculus)

Let C be a positivley oriented, piecwise-smooth, simple closed curve int he plance and let D be the region bounded by C. If P and Q have contionous partial derivatives on an open region that contains D, then

$\displaystyle \oint_C Pdx+Qdy=\iint_D \left( \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dA$

You started with

$\displaystyle \int_c (\sigma_y dx - \sigma_x dy)$

note that $\displaystyle P=\sigma_y \mbox{ and } Q=-\sigma_x$

Now using Greens theorem

$\displaystyle \int_c (\sigma_y dx - \sigma_x dy) =\iint_D -\sigma_{xx}-\sigma_{yy} dA=-\iint \sigma_{xx}+\sigma_{yy}dA=-\iint 0 dA=0$