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→More About Coordinate Systems: Add more.
And, going the other way:
And, going the other way:
::<math>\begin{bmatrix} A_x \\ A_y \end{bmatrix} = \begin{bmatrix} \cos \theta & - r \sin \theta \\ \sin \theta & r \cos \theta \end{bmatrix} \begin{bmatrix} A_r \\ A_\theta \end{bmatrix}</math>
::<math>\begin{bmatrix} A_x \\ A_y \end{bmatrix} = \begin{bmatrix} \cos \theta & - r \sin \theta \\ \sin \theta & r \cos \theta \end{bmatrix} \begin{bmatrix} A_r \\ A_\theta \end{bmatrix}</math>
Let's look at some vectors whose components are given in general coordinates—there are some important and striking aspects of this.
In polar coordinates, a "radial basis vector" at any point has components:
::<math>(A_r=1, A_\theta=0)</math>
Its Cartesian components (from the formulas above) are:
::<math>(A_x=\cos \theta, A_y=\sin \theta)</math>
The actual (Cartesian) direction in which the vector points depends on its location. It always points radially outward. Its length is <math>\sqrt{A_x^2 + A_y^2} = \sqrt{\cos^2 \theta + \sin^2 \theta} = 1</math>. So the radial basis vector is a unit vector.
The "angular basis vector" has components:
::<math>(A_r=0, A_\theta=1)</math>
Its Cartesian components are:
::<math>(A_x=- r \sin \theta, A_y=r \cos \theta)</math>
It always points "laterally", at right angles to the radial direction. More surprisingly, its length is <math>\sqrt{A_x^2 + A_y^2} = r</math>. The angular basis vector is not a unit vector! It is bigger when farther away from the origin, even though the sum of the squares of its components is 1.<ref>Some textbooks use a definition of vector components such that basis vectors always have length 1. Such a treatment imposes the requirement that the coordinate lines be perpendicular everywhere. The formulas for vector operations are more complicated and less elegant when this is done.</ref>
We need a better formula for the dot product than the rule that works for Cartesian coordinates:
::<math>\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z</math>
===The Dot Product in Arbitrary Coordinates===
===The Cross Product in Arbitrary Coordinates===
=Introduction to Integrals in Multivariable Calculus=
=Introduction to Integrals in Multivariable Calculus=