- #1
Markus Hanke
- 259
- 45
Is there a geometric interpretation/visualization of the exterior derivative, at least in the case of three dimensions ?
Suppose we have a 1-form on a 3-dimensional basis {dx1, dx2,dx3} :
[tex]\displaystyle{\omega =f_{i}dx^{i}}[/tex]
with a set of real-valued coefficients f. The exterior derivative is then, by definition, the 2-form
[tex]\displaystyle{d\omega =\sum_{i,j}\frac{\partial f_{j}}{\partial x^{i}}dx^{i}\wedge dx^{j}}[/tex]
Intuitively, a 1-form in three dimensions is an oriented line segment, a 2-form an oriented surface element. So, is there an intuitive way to "visualize" the exterior derivative operation, in a geometric sense ? Can it be visualised roughly as "wedging" with an orthogonal basis element in a way that orientation is preserved, thereby increasing the degree by one ? This would explain how, for example, the exterior derivative turns an oriented line segment into an oriented surface element.
I am fine with the abstract definitions of the operators, and its connections to the usual div/grad/curl, but it would be very helpful to have some way to intuitively visualise it as well. Ultimately I am trying to gain an intuitive understanding of the differential forms notation for the Maxwell equations; my problem is that, just by looking at dF=0 and d*F=uJ it is very hard to visualise what this actually implies in a geometric sense.
Suppose we have a 1-form on a 3-dimensional basis {dx1, dx2,dx3} :
[tex]\displaystyle{\omega =f_{i}dx^{i}}[/tex]
with a set of real-valued coefficients f. The exterior derivative is then, by definition, the 2-form
[tex]\displaystyle{d\omega =\sum_{i,j}\frac{\partial f_{j}}{\partial x^{i}}dx^{i}\wedge dx^{j}}[/tex]
Intuitively, a 1-form in three dimensions is an oriented line segment, a 2-form an oriented surface element. So, is there an intuitive way to "visualize" the exterior derivative operation, in a geometric sense ? Can it be visualised roughly as "wedging" with an orthogonal basis element in a way that orientation is preserved, thereby increasing the degree by one ? This would explain how, for example, the exterior derivative turns an oriented line segment into an oriented surface element.
I am fine with the abstract definitions of the operators, and its connections to the usual div/grad/curl, but it would be very helpful to have some way to intuitively visualise it as well. Ultimately I am trying to gain an intuitive understanding of the differential forms notation for the Maxwell equations; my problem is that, just by looking at dF=0 and d*F=uJ it is very hard to visualise what this actually implies in a geometric sense.