Changes

===The Dot Product in Arbitrary Coordinates===

===The Dot Product in Arbitrary Coordinates===

Given two vectors <math>\vec{A}</math> and <math>\vec{B}</math>, described in a u/v/w coordinate system, we can find their dot product by converting to Cartesian coordinates and using the simple rule that we know, which is the sum of the pairwise products of the vector components.  This means that, at least momentarily, we will have to keep track of the vectors' components in both the Cartesian system, which we have been calling <math>A_x</math>, <math>A_y</math>, and <math>A_z</math>, and in the new coodinate system, which we have been calling <math>A_u</math>, <math>A_v</math>, and <math>A_w</math>.  To do the required matrix manipulations, we have to give the components numbers instead of names.  Things like <math>A_{u_1}</math>, <math>A_{u_2}</math>, and <math>A_{u_3}</math> are just too unwieldy.<ref>This is a well known notational problem in multivariable consequence.  It is one of the reasons why tensor algebra textbooks are so hard to read.</ref>  So we will make this simplification:

Given two vectors <math>\vec{A}</math> and <math>\vec{B}</math>, described in a u/v/w coordinate system, we can find their dot product by converting to Cartesian coordinates and using the simple rule that we know, which is the sum of the pairwise products of the vector components.  This means that, at least momentarily, we will have to keep track of the vectors' components in both the Cartesian system, which we have been calling <math>A_x</math>, <math>A_y</math>, and <math>A_z</math>, and in the new coodinate system, which we have been calling <math>A_u</math>, <math>A_v</math>, and <math>A_w</math>.  To do the required matrix manipulations, we have to give the components numbers instead of names.  Things like <math>A_{u_1}</math>, <math>A_{u_2}</math>, and <math>A_{u_3}</math> are just too unwieldy.<ref>This is a well known notational problem in multivariable calculus.  It is one of the reasons why tensor algebra textbooks are so hard to read.</ref>  So we will make this simplification:

*The components of a vector in the new coordinate system (formerly <math>A_u</math>, <math>A_v</math>, and <math>A_w</math>) will just be called <math>A_1</math>, <math>A_2</math>, and <math>A_3</math>.

*The components of a vector in the new coordinate system (formerly <math>A_u</math>, <math>A_v</math>, and <math>A_w</math>) will just be called <math>A_1</math>, <math>A_2</math>, and <math>A_3</math>.

*The components of a vector in the "reference" Cartesian system will be written with a tilde: <math>\tilde{A_1}</math>, <math>\tilde{A_2}</math>, and <math>\tilde{A_3}</math>.

*The components of a vector in the "reference" Cartesian system will be written with a tilde: <math>\tilde{A_1}</math>, <math>\tilde{A_2}</math>, and <math>\tilde{A_3}</math>.