Integrating a 2-form on a 2-cube

[quasar987]

I read, again in Spivak's Calculus on Manifolds, that the integral of 1-form over a 1-cube is equivalent to a line integral. And indeed, if I consider the 1-form w = Pdx + Qdy on R², and c a given 1-cube in R², I find that

$$\int_c\omega = \int_0^1 F(c(t))\cdot c'(t)dt$$

where F=(P,Q), which is the integral of the vector field F along the curve c.

Next, I read that the integral of a 2-form over a 2-cube is equivalent to a surface integral. Let $\omega = Pdx\wedge dy + Qdx\wedge dz + Rdy\wedge dz$ be a 2-form onR³ and c be a 2-cube in R³. I am guessing that $\int_c\omega$ corresponds to the flux integral of the vector field F = (P,Q,R) through c([0,1]²). Recall that this integral is given by

$$\iint_{[0,1]^2}F(c(s,t))\cdot \left(\frac{\partial c}{\partial s}\times \frac{\partial c}{\partial t}\right) dsdt=\iint_{[0,1]^2}P(c(s,t))\left| \begin{array}{cc} \frac{\partial c^2}{\partial s} & \frac{\partial c^3}{\partial s} \\ \frac{\partial c^2}{\partial t} & \frac{\partial c^3}{\partial t} \end{array} \right| - Q(c(s,t))\left| \begin{array}{cc} \frac{\partial c^1}{\partial s} & \frac{\partial c^3}{\partial s} \\ \frac{\partial c^1}{\partial t} & \frac{\partial c^3}{\partial t} \end{array} \right| + R(c(s,t))\left| \begin{array}{cc} \frac{\partial c^1}{\partial s} & \frac{\partial c^2}{\partial s} \\ \frac{\partial c^1}{\partial t} & \frac{\partial c^2}{\partial t} \end{array} \right| \ dsdt$$

Alright, so I get something close to this, but not quite, so I'm wondering where I'm mistaken... I say,

$$\int_c\omega = \int_{[0,1]^2}(P\ \circ \ c) c^*(dx\wedge dy)+(Q\ \circ \ c) c^*(dx\wedge dz)+(R\ \circ \ c) c^*(dy\wedge dz)$$

Now, $c^*(dx^i\wedge dx^j))$ is a 2-form on [0,1]² so it is of the form $f_{ij} \ ds\wedge dt$ for some 0-form f_ij on [0,1]². In order to find f_ij, let p belong to [0,1]² and $v_{p}, \ w_{p} \in \mathbb{R}^2_{p}$.

Using the definitions, I find that

$$c^*(dx^i\wedge dx^j)(v_p,w_p)=dx^i\wedge dx^j(c_*(v_p),c_*(w_p))=dx^i\otimes dx^j(c_*(v_p),c_*(w_p))-dx^i\otimes dx^j(c_*(w_p),c_*(v_p))$$

where $c_*(v_p)=(Dc(p)(v))_{c(p)}$, and therefor

$$dx^i(c_*(v_p))=\frac{\partial c^i}{\partial s}(p)v^1+\frac{\partial c^i}{\partial t}(p)v^2=\left(\frac{\partial c^i}{\partial s}(p)ds+\frac{\partial c^i}{\partial t}(p)dt\right)(v_p)$$

So I conclude that

$$c^*(dx^i\wedge dx^j)(v_p,w_p)=\left(\frac{\partial c^i}{\partial s}(p)dx+\frac{\partial c^i}{\partial t}(p)dt\right)\wedge \left(\frac{\partial c^j}{\partial s}(p)dx+\frac{\partial c^j}{\partial t}(p)dt\right)=\left(\frac{\partial c^i}{\partial s}(p)\frac{\partial c^j}{\partial t}(p)-\frac{\partial c^i}{\partial t}(p)\frac{\partial c^j}{\partial s}(p)\right) ds\wedge dt(v_p,w_p)$$

and hence

$$f_{ij}=\frac{\partial c^i}{\partial s}\frac{\partial c^j}{\partial t}-\frac{\partial c^i}{\partial t}\frac{\partial c^j}{\partial s}=\left| \begin{array}{cc} \frac{\partial c^i}{\partial s} & \frac{\partial c^j}{\partial s} \\ \frac{\partial c^i}{\partial t} & \frac{\partial c^j}{\partial t} \end{array} \right|$$

Finally,

$$\int_c\omega = \iint_{[0,1]^2}P(c(s,t))\left| \begin{array}{cc} \frac{\partial c^1}{\partial s} & \frac{\partial c^2}{\partial s} \\ \frac{\partial c^1}{\partial t} & \frac{\partial c^2}{\partial t} \end{array} \right| + Q(c(s,t))\left| \begin{array}{cc} \frac{\partial c^1}{\partial s} & \frac{\partial c^3}{\partial s} \\ \frac{\partial c^1}{\partial t} & \frac{\partial c^3}{\partial t} \end{array} \right| + R(c(s,t))\left| \begin{array}{cc} \frac{\partial c^2}{\partial s} & \frac{\partial c^3}{\partial s} \\ \frac{\partial c^2}{\partial t} & \frac{\partial c^3}{\partial t} \end{array} \right| \ dsdt$$

So, 3 things differ: the factors of P and R and the sign of Q.

Thanks!

 

[Doodle Bob]

Next, I read that the integral of a 2-form over a 2-cube is equivalent to a surface integral. Let $\omega = Pdx\wedge dy + Qdx\wedge dz + Rdy\wedge dz$ be a 2-form onR³ and c be a 2-cube in R³. I am guessing that $\int_c\omega$ corresponds to the flux integral of the vector field F = (P,Q,R) through c([0,1]²). ...

So, 3 things differ: the factors of P and R and the sign of Q.

Thanks!

I'm not seeing any problem here. Modulo any sign errors that I didn't pick up, what you've shown is that the integral of a 2-form on a 2-chain in 3-space $\int_c\omega$ is the flux integral of the vector field F=(R,-Q,P), where P, Q and R are the coordinate components of the 2-form.

Integrating forms on chains is a generalization of curve and surface integration, but it's not necessarily a direct correspondence. Some adjustment of the various components usually has to be performed.

Incidentally, calculus on manifolds is not Spivak's invention. His goal with this book was to provide a short, problem-based introduction to the subject. A great companion book to his is Loomis and Sternberg's Advanced Calculus, which you can get for free here: http://www.math.harvard.edu/~shlomo/.

 

[quasar987]

Ok, sweet! And thanks for the book link.