Math for Electricity and Magnetism

The magnitude of the vector product equals the area of the parallelogram spanned by vectors a and b, |a| |b| sin φ, with a unit vector perpendicular to the plane.

Study the triple scalar product a · (b × c); it vanishes for parallel vectors or when a is perpendicular to b × c, and swapping rows flips the sign.

Explore the cylindrical basis in electromagnetism: define the local unit vectors e_rho, e_phi, and e_z, relate them to Cartesian units, and visualize surfaces of constant rho, phi, and z.

Relate cartesian coordinates to cylindrical coordinates using x = r cos phi, y = r sin phi, with r = sqrt(x^2 + y^2), phi = arctan(y/x), and z remains unchanged.

We express e_z in the cartesian basis using the second method; with phi constant, the partial derivative with respect to z is zero, yielding e_z as a unit vector.

Explore rotation matrices in three dimensions, showing that the transpose equals the inverse for 3×3 matrices and how identity relationships arise from a three-by-three basis.

Examine spherical coordinates, linking the radius r and angles theta and phi to a point in three-dimensional space, and clarify the range and interpretation of r, theta, and phi.

Examine how three mutually orthogonal unit vectors e_r, e_theta, and e_phi form a local spherical coordinate basis, with the position vector r e_r as a function of theta and phi.

Explain expressing the position vector in Cartesian form x i + y j + z k and in spherical form as r e_r with unit vectors e_r, e_theta, e_phi.

Derive how the spherical unit vectors r̂, θ̂, and φ̂ express in the Cartesian basis Ex, Ey, and k̂, using cosφ and sinφ relations.

Remark 3 explains how cosigner states combine with pi over two shifts to yield sinus state or cosigner state, showing sign relations that map between these states.

Define the infinitesimal displacement vector in Cartesian coordinates by combining dx, dy, dz into ds, and relate it to infinitesimal line and area elements.

Explore the area element for the cylinder’s lateral surface using polar coordinates, deriving the small arc and differential element ds with radius r, phi, and z.

Explain the infinitesimal displacement vector in spherical coordinates, deriving dl = e_r dr + r de_r and showing how theta and phi derivatives of r e_r determine the displacement.

Explore how the divergence operator acts on a vector field to yield a scalar, using Cartesian form ∂A_x/∂x + ∂A_y/∂y + ∂A_z/∂z and spherical components A_r, A_θ, A_φ.