Area of Region Vector Calculus

[i_hate_math]

I have tried to apply greens theorem with P(x,y)=-y and Q(x,y)=x, and gotten ∫ F • ds = 2*Area(D), where F(x,y)=(P,Q) ===> Area(D) = 1/2 ∫ F • ds = 1/2 ∫ (-y,x) • n ds . This is pretty much the most common approach to an area of region problem. But here they ask you to prove this bizarre relation of Area(D) = 1/2 ∫ F • ds = ∫ (x,y) • n ds. I am clueless what to do.

Without a good understanding of part (a) of the question, I don't know how to approach (b) at all. I know the parameterisation could be x=acost , y=bsint, 0≤b≤2π. It seems easy but I am in desperate need of some guidance.

Thanks heaps for helping!

 

[Ray Vickson]

I have tried to apply greens theorem with P(x,y)=-y and Q(x,y)=x, and gotten ∫ F • ds = 2*Area(D), where F(x,y)=(P,Q) ===> Area(D) = 1/2 ∫ F • ds = 1/2 ∫ (-y,x) • n ds . This is pretty much the most common approach to an area of region problem. But here they ask you to prove this bizarre relation of Area(D) = 1/2 ∫ F • ds = ∫ (x,y) • n ds. I am clueless what to do.

Without a good understanding of part (a) of the question, I don't know how to approach (b) at all. I know the parameterisation could be x=acost , y=bsint, 0≤b≤2π. It seems easy but I am in desperate need of some guidance.

Thanks heaps for helping!

For a very short line segment from $(x_0,y_0)$ to $(x_0 + \Delta x, y_0 + \Delta y)$ on the curve $C$, what would be the normal $\vec{n}$ in terms of $x_0, y_0, \Delta x, \Delta y$? If $\Delta s$ is the distance from $(x_0,y_0)$ to $(x_0 + \Delta x, y_0 + \Delta y)$, what would be the value of $(x,y) \cdot \vec{n} \Delta s$? If the origin (0,0) is in the interior of the region $D$, what would you get if you summed over all those small increments like those you just computed above?

Finally, what happens if (0,0) is exterior to $D$?

 

[i_hate_math]

For a very short line segment from $(x_0,y_0)$ to $(x_0 + \Delta x, y_0 + \Delta y)$ on the curve $C$, what would be the normal $\vec{n}$ in terms of $x_0, y_0, \Delta x, \Delta y$? If $\Delta s$ is the distance from $(x_0,y_0)$ to $(x_0 + \Delta x, y_0 + \Delta y)$, what would be the value of $(x,y) \cdot \vec{n} \Delta s$? If the origin (0,0) is in the interior of the region $D$, what would you get if you summed over all those small increments like those you just computed above?

Finally, what happens if (0,0) is exterior to $D$?

n = (dy/ds, −dx/ds) I guess?

 

[Ray Vickson]

n = (dy/ds, −dx/ds) I guess?

Why not just $\vec{n} = (\Delta y, -\Delta dx) / \sqrt{\Delta x^2 + \Delta y^2}$? (That is, if you want a unit normal!)

 

[i_hate_math]

I think ds is equivalent to sqrt(dx +dy)?

 

[Ray Vickson]

I think ds is equivalent to sqrt(dx +dy)?

No; it is $\sqrt{dx^2+dy^2}$.