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Terilien
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What exactly is the exterior derivative? What is its motivation? how do you compute it? Most importantly why is that how you copute it?
Well, if you followed mathwonk's description that we use the exterior derivative because it makes Stoke's theorem work... then apply Stoke's theorem!Terilien said:Well could someone explain why we impose the condition d(da)=0. I think I understand but would still like an explanation...
Terilien said:Well could someone explain why we impose the condition d(da)=0, and it's not something to be "imposed" but is instead a consequence of the manner in which the exterior derivative is defined. I think I understand but would still like an explanation...
Terilien said:So I did understand it. One last thing, I'm not sure if I understand the modified leibniz rule very well. could someone prove it rigorously?
I'm talking about the liebniz rule between wedge products. I don't quite know how to prove it...:(
When doing exterior algebra, I'm very much used to writing the product asTerilien said:I was talking about the exterior derivative of a wedge product.
It's suppose to be something like, p^dq +-1^p (q^dp) or something along those lines. how do we get that?>
i know its silly but i really don't know how its proven.
That (generally) shouldn't be equal to d(p /\ q). So it's a good thing you didn't wind up with p /\ dq - q /\ dp.Terilien said:what i did was (p+dp)^(q+dq) -p^q. i evaluated that and got + p^dq +dp^q +dp^dq
what do we do with that?
Well, I'm not really sure what you're asking anymore. And it's past my bedtime, so I can't help anymore today.Terilien said:so where does the other thing come into play?
Terilien said:What exactly is the exterior derivative? What is its motivation? how do you compute it? Most importantly why is that how you copute it?
Terilien said:I was talking about the exterior derivative of a wedge product.
It's suppose to be something like, p^dq +-1^p (q^dp) or something along those lines. how do we get that?>
i know its silly but i really don't know how its proven.
The exterior derivative is a mathematical operation that is used to measure how much a function changes over a surface or in a higher-dimensional space. It is a tool commonly used in differential geometry and calculus to understand the behavior of functions on manifolds.
The exterior derivative is calculated by taking the partial derivatives of a function with respect to each variable, and then combining them using the wedge product to create a differential form. This process can be generalized to higher dimensions, where the exterior derivative is represented by the exterior derivative operator, d.
The exterior derivative is motivated by the concept of differential forms, which are mathematical objects that can be used to describe the behavior of functions on manifolds. By using the exterior derivative, we can gain insight into the geometry and topology of a manifold by studying the behavior of functions on it.
The exterior derivative is closely related to the gradient, divergence, and curl operations in multivariable calculus. In fact, the exterior derivative can be seen as a generalization of these operations to higher dimensions. Each of these operations can be obtained from the exterior derivative by manipulating the differential forms that it produces.
The exterior derivative has many applications, particularly in physics and engineering. It is used to describe the behavior of electromagnetic fields in Maxwell's equations, as well as in the study of fluid mechanics and general relativity. It is also essential in understanding the concept of integration on manifolds, which is used in fields such as computer graphics and computer-aided design.