Recent content by xio

If fact, at least one does -- Apostol's "Calculus: II", page 434, section 12.9 "Other notations for surface integrals": first the relation between the direction of the normal and its sign is discussed and then the following formula is introduced: $$\begin{eqnarray*}...

Hurray! Vela, LCKurtz, thanks a lot!

In our case $$\mathbf{n}=\begin{vmatrix}\boldsymbol{\hat{ \imath}} & \boldsymbol{\hat{\jmath}} & \boldsymbol{\hat{k}}\\ 1 & 1 & -2\\ 1 & -1 & 0 \end{vmatrix}=(-2,-2,-2) , $$ but since its ##z##-component is negative we take ##-\mathbf{n}=(2,2,2)##; is this correct?

So there can be two normals to the surface, each in the opposite direction: $$\mathbf{n}_{1}=\frac{\partial\mathbf{r}}{ \partial u}\times\frac{\partial\mathbf{r}}{\partial v}/\left\Vert \frac{\partial\mathbf{r}}{\partial u}\times\frac{\partial\mathbf{r}}{\partial v}\right\Vert...

Hrm... the book says additionally: "Let ##\mathbf{n}## denote the unit normal to ##S## having a nonnegative ##z##-component", but I cannot see how it can help right now...

Vela, LCKurtz, so I map my xy-triangle onto the uv-plane: and evaluate the integral accordingly: $$\iint_{S}\mathbf{F}\cdot\mathbf{n}\, dS=-2\iint_{T}\, du\, dv=-2\int_{0}^{0.5}\left[\int_{-u}^{u}dv\right]du=-\frac{1}{2}. $$ However, as you may see I get a negative value. Is this what is...

[EDIT]: Correct answer for this problem is 1/2, not 4 as I thought before; that means the result for the explicit representation was correct. Still I don't understand how to treat the case with the parametric representation. Greetings, I need to evaluate $$\iint_{S}\mathbf{F}\cdot\mathbf{n}\...

I never suggested that. Change of variables is some ten subsections below my current position in Apostol, but I liked the technique, thank you.

Hrm... I guess I cheated: I took the total volume to be the sum of volumes over the triangles formed by the axes and four diagonal lines (##x+1##, ##1-x##, ##-x-1##, ##x-1##): $$ \int_{-1}^{0}\left[\int_{0}^{x+1}{\rm e}^{x+y}dy\right]dx+\int_{-1}^{0}\left[\int_{-x-1}^{0}{\rm...

Oh Lord, ##{\rm e}^{x+y}## obviously isn't symmetric.

[EDIT]: Found the mistake, see the next post. Homework Statement Evaluate $$\iint_{S}{\rm e}^{x+y}dx\, dy,S=\{(x,y):\left|x\right|+\left|y\right|\leq1\} $$ 2. The attempt at a solution ##\left|x\right|+\left|y\right|## is the rhombus with the center at the origin, symmetrical about both...