## Integrating a p - form

Let
$\omega_p$
be a differential p - form in
$\mathbb{R}^n$
and let
$K$
be a closed bounded rectangle in
$\mathbb{R}^p$

Let
$f : \mathbb{R}^p \rightarrow \mathbb{R}^n$
be differentiable. In the diagram below,
$f(K)=S$

If
$\mathbf{u}=(u_1 , ..., u_p) \in K$
then
$f\mathbf{(}\mathbf{u})=(f_1(\mathbf{u}),...,f_n (\mathbf{u})) \in S$

We can apply
$\omega^p$
at a point
$\mathbf{f}(\mathbf{u})$
to the set
$\frac{\partial \mathbf{f}}{\partial u_1} \mathbf{u}, ..., \frac{\partial \mathbf{f}}{\partial u_p} \mathbf{u}$
t obtain
$\omega^p_{\mathbf{f}(\mathbf{u})}(\frac{\partial \mathbf{f}}{\partial u_1} \mathbf{u}, ..., \frac{\partial \mathbf{f}}{\partial u_p} \mathbf{u})$

If
$\Omega$
is a grid cover of
$K$
with vertices
$(\mathbf{u}_1 ,..., \mathbf{u}_\alpha )$
and rectangles
$K_1,..., K_\alpha$
we can form the sum
$\sum_{i=1}^\alpha \omega^p_{\mathbf{f}(\mathbf{u})}(\frac{\partial \mathbf{f}}{\partial u_1} \mathbf{u}, ..., \frac{\partial \mathbf{f}}{\partial u_p} \mathbf{u}) V(K_i)$

where
$V(K_i)$
is the p - dimensional volume of
$K_i$

The integral of
$\omega^p$
over
$S$
is
$\int_S \omega^p = lim_{D(\Omega) \rightarrow 0} \sum_{i=1}^\alpha \omega^p_{\mathbf{f}(\mathbf{u})}(\frac{\partial \mathbf{f}}{\partial u_1} \mathbf{u}, ..., \frac{\partial \mathbf{f}}{\partial u_p} \mathbf{u}) V(K_i)$

The sum on the right hand side is a Riemann sum, and if it exists then
$\int_S \omega^p = \int_K \omega_{\mathbf{f} ( \mathbf{u})} (\frac{\partial \mathbf{f}}{\partial u_1} ,..., \frac{\partial \mathbf{f}}{\partial u_p} ) dV$
(1)
If p=1 then
$K=[a,b]$
and if
$t \in [a,b]$
then
$\mathbf{f}(t) \in S \subset \mathbb{R}^n$
.
$\mathbf{f}(t)$
will be a curve in
$\mathbb{R}^n$

If
$\omega'{\mathbf{x}} =F_1 (\mathbf{x})dx_1 +...+F_n (\mathbf{x})dx_n$

then (1) reduces to
\begin{aligned} \int_S \omega'_{\mathbf{f}(t)} &= \int^b_a \omega'_{\mathbf{f}(t)} (\frac{\partial \mathbf{f}}{\partial t}) dt \\ &= \int^b_a (F_1 (\mathbf{x}) \frac{\partial f_1}{dt} +...+F_n (\mathbf{x}) \frac{\partial f_n}{dt}) dt \end{aligned}