Talk:Vector calculus

Parallel wikiversity efforts

(title later added by --guyvan52 (discuss • contribs) 18:03, 2 February 2014 (UTC)) Is this page completely necessary in that there already exists a Vector page([[1]]) which has a decent vector calc section within it. I see two options, the long term and the short term. For the short term we can keep the vector page and continue to edit and add to it (its fine for a page overviewing vectors but it currently has too little to stand by itself since each definition is short and too specific). Once the vector calc sections become large enough they can be moved together into the vector calc page and shorter entries can be replaced once again. I guess I should not have said two options since I suggest only one path. What do other people think about this plan?

Yes, I just looked at the vector page an it looks pretty good. Your "wait and see" approach is the only logical one. I will now write a short introduction to this Vector calculus page so you get some insight as to where we hope to go with this page. You should know that the name of this page is not entirely appropriate, but that I don't know exactly what to name it. It is intended to be a short and simple introduction to both the Riemannian calculus used in General Relativity, and also the dual non-orthonormal spaces associated with solid state physics. But, it will contain little if any actual physics.--guyvan52 (discuss • contribs) 01:55, 29 January 2014 (UTC)

Tentative outline

I am moving this to the talk page because this outline is very long and not of interest to most readers:

The study of vector calculus in two dimensions permits us to quickly prove theorems that can be generalized to higher dimensions. It is often the case that proof of these higher dimensional generalizations are too tedious to be pedagogically useful. By studying the two dimensional proof the student can get the general idea of how the proof in higher dimensions might proceed. --guyvan52 (discuss • contribs) 17:41, 2 February 2014 (UTC)

By focusing on two dimensions, this work will not fully cover the curl, gradient, and divergence, as it is usually applied to a field such as electromagnetism or classical dynamics. In contrast, this work will focus on systems that are not orthonormal. Topics to be introduced include:

1. Rotation of coordinates (including a derivation of the rotation matrix)
2. Proof that the following entities are invarient under rotations:
   1. Inner product
   2. Gradient of a scalar field
   3. Laplacian of a scalar field
   4. Counter example: Why $\partial ^{2}\Phi /\partial x\partial y$ is NOT symmetric under rotations.
3. A two dimensional lattice structure as an example of a non-orthonormal coordinate system.
   1. The need for a dual basis in order to compactly write the inner product
   2. Proof that the inner product defined within this dual basis is also symmetric under rotations.
   3. Interpretation of the dual basis in terms of waves that will exhibit Bragg scattering of this crystal structure. To give the student a glimpse of how solid state physics uses these dual basis vectors (r-space and k-space), the wavefronts associated with waves that do and do not exhibit Bragg scattering will be superimposed over the lattice structure.
4. Introduction to the tensor field as a generalized spring constant. A figure showing a particle held by four orthogonal springs will be shown. The tensor becomes non-diagonal if the springs do not align with the x and y axis. To simplify the notation, this will be case in terms of a force and displacement, although in most applications the tensor refers to polarization response.
   1. Invariance of this tensor under rotations will be proven
   2. Students have difficulty visualizing the tensor field. The concept of a tensor field is introduced by assuming that a crystal has weak inhomogeniety in the nature of these springs mechanisms as one travels distances much larger than the size of the lattice structure.
5. The two-dimensional analog of the divergence theorem and Stokes theorem will be stated. The latter is called Green's theorem, but it will be stated by introducing the curl operator that maps a 2D space into a 1 D space (typically associated with the ${\hat {z}}$ direction.)
6. A two dimensional version of a scalar electrostatic field is introduced. The "field" will fall as 1/r and the potential falls logarithetially.
   1. This field theory will be depicted in the various notations used in physics, from partial differential equatiion.
   2. The potential form of Maxwell's equations will be introduced (using ${\vec {A}},\Phi$ instead of ${\vec {E}},{\vec {B}}$)
7. Noneuclidean calculus will be introduced with a metric that will eventually be shown to that of a parabola and/or sphere.
   1. A hyperbolic metric will also be introduced
   2. Riemannian curvature will be introduced, which is a scalar in a two dimensional space.
8. This will eventually lead us to curvilinear coordinates and an attempt to contruct linearized general relativity --guyvan52 (discuss • contribs) 17:41, 2 February 2014 (UTC)

Active and passive rotations

In the active (alibi) transformation (left), a point moves from position P to P' by rotating clockwise by an angle θ about the origin of the coordinate system. In the passive (alias) transformation (right), point P does not move, while the coordinate system rotates counterclockwise by an angle θ about its origin. The coordinates of P' in the active case are the same as the coordinates of P relative to the rotated coordinate system.

We shouldn't focus on this topic, but the figure is helpfull in allowing us to properly use the terms. The figure can be found by linking to w:Active and passive transformation

--guyvan52 (discuss • contribs) 18:01, 2 February 2014 (UTC)