Hey /sci/ does anyone have a good intuition what differential

>>8491786

" muh intuition"

It's just PDE, [math] df = xdy + ydx [/math] or smt like that. If you do not get the definition, make some exercices.

>>8491786
The differential form of a function is kinda like the rate at which said function varies. At least, that's how I 'intuitively' think of it. Just keep in mind that this is rather vague and the actual definition is how you should look at it.

>>8491786
Draw a picture. The definitions describe the geometry. Don't start with definitions unless you know what they look like.

>>8491786
Well a differential form is a map that associates to each point on a manifold a k-linear alternating form on the tangent space.
So first, we need to think about alternating multilinear forms. A great example of such a form on a vector space V of dimension n, with a basis [math]B[/math], is the determinant [math]\det_B[/math], which associates to n vectors the oriented volume (relative to B) of the parallelotope generated by those vectors.
Now, of course, it is not so easy to understand k-linear alternating forms on a vector space of dimension n for k < n and what they measure.
The point I want to make is that they still measure "the" k-volume of a k-parallelotope, only that because k < n, there are many (independent) ways to do it.
What I mean is that the volume of a parallelotope generated by k vectors [math]u_1, \dots, u_k[/math] should be thought of as the wedge product [math]u_1 \wedge \dots \wedge u_k[/math] (which, as you notice, is intrinsic) and that each linear form [math]\phi \in \left(\Lambda^k V\right)^*[/math] is a way of measuring volumes (it's a way of interpreting the property that k-linear alternating maps factor through the k-th exterior power)
When n = k, there is only one such way to measuring, up to a constant scaling factor (corresponding to a choice of orientation and units, it's the example of the determinant with respect to a basis that we discussed above), but when k < n, there are several: For example, if you choose a basis [math]v_1, \dots v_n[/math], you can define such a measure as the volume of the projection onto [math]\langle v_{i_1}, \dots, v_{i_k} \rangle[/math] along [math]\langle v_j, j \not \in \{i_1,\dots i_k\}\rangle[/math] for each i1 < ... < ik.
Now finally, what is a differential k-form ? It's a way to coherently assign a weighted volume to each tangent k-parallelotope (that tangent parallelotope represents an "infinitesimal element of k-volume").

>>8491786
A differential k-form on a manifold X is a smooth section of the kth exterior power of the cotangent bundle of X.

http://www.dummies.com/education/math/calculus/how-to-analyze-position-velocity-and-acceleration-with-differentiation/