Comments for The Unapologetic Mathematician

Comment on Vector-Valued Functions by The Jacobian « The Unapologetic Mathematician

The Jacobian « The Unapologetic Mathematician — Wed, 11 Nov 2009 17:32:47 +0000

[...] some open region in . That is, if we pick a basis and coordinates of , then the function is a vector-valued function of real variables with components . The differential, then, is itself a vector-valued function [...]

Comment on Calculating the Determinant by The Jacobian « The Unapologetic Mathematician

The Jacobian « The Unapologetic Mathematician — Wed, 11 Nov 2009 16:50:06 +0000

[...] to as the Jacobian, or the Jacobian matrix. Since this matrix is square, we can calculate its determinant, which is also referred to as the Jacobian, or the Jacobian determinant. I’ll try to be clear [...]

Comment on Euclidean Spaces by The Jacobian « The Unapologetic Mathematician

The Jacobian « The Unapologetic Mathematician — Wed, 11 Nov 2009 16:49:07 +0000

[...] rule, the differential at the point defines a linear transformation from the -dimensional space of displacement vectors at to the -dimensional space of displacement vectors at , and the matrix entries with respect to [...]

Comment on The Chain Rule by The Jacobian « The Unapologetic Mathematician

The Jacobian « The Unapologetic Mathematician — Wed, 11 Nov 2009 16:48:07 +0000

[...] like we said when discussing the chain rule, the differential at the point defines a linear transformation from the -dimensional space of [...]

Comment on Differentials by The Jacobian « The Unapologetic Mathematician

The Jacobian « The Unapologetic Mathematician — Wed, 11 Nov 2009 16:46:50 +0000

[...] function of real variables with components . The differential, then, is itself a vector-valued function whose components are the differentials of the component [...]

Comment on Parallelepipeds and Volumes I by The Jacobian « The Unapologetic Mathematician

The Jacobian « The Unapologetic Mathematician — Wed, 11 Nov 2009 16:45:34 +0000

[...] Now that we’ve used exterior algebras to come to terms with parallelepipeds and their transformations, let’s come back to apply these ideas to the [...]

Comment on Exterior Algebras by The Jacobian « The Unapologetic Mathematician

The Jacobian « The Unapologetic Mathematician — Wed, 11 Nov 2009 16:44:29 +0000

[...] Now that we’ve used exterior algebras to come to terms with parallelepipeds and their transformations, let’s come back to apply [...]

Comment on The Cross Product and Pseudovectors by Qiaochu Yuan

Qiaochu Yuan — Tue, 10 Nov 2009 17:44:13 +0000

Thank you! I’ve always wanted to see a clear explanation of this online, and I think the Wikipedia articles are a little too intimidating.

Comment on Antisymmetric Tensors by The Cross Product and Pseudovectors « The Unapologetic Mathematician

The Cross Product and Pseudovectors « The Unapologetic Mathematician — Tue, 10 Nov 2009 17:02:12 +0000

[...] we can see exactly what’s going on. These are just the spaces , , , and , along with their representations of the orthogonal group . And the “pseudo” means we’ve used the Hodge star [...]

Comment on Orthogonal transformations by The Cross Product and Pseudovectors « The Unapologetic Mathematician

The Cross Product and Pseudovectors « The Unapologetic Mathematician — Tue, 10 Nov 2009 17:01:05 +0000

[...] and inner product-preserving transformations, but we can also throw in reflections to get the whole orthogonal group, of all transformations from one orthonormal basis to [...]