Chapter 1: Theory of Vector Field

• Laws of E&M expressed in terms of E and B fields
• Differential eq. involve vector derivatives

If \(\nabla \cdot \boldsymbol{F} = D\) and \(\nabla \times \boldsymbol{F} = \boldsymbol{C} \rightarrow \nabla \cdot \boldsymbol{C} = 0\)
Can you find \(\boldsymbol{F}\) given \(D\) and \(\boldsymbol{C}\) (divergence and curl of \(\boldsymbol{F}\))?
Yes, given appropriate boundary conditions.
Helmholtz Theorem guarantees that field is uniquely determined by its divergence and curl, with known boundary conditions.

Theorem 1: Curl-less (irrotational) fields

1. \(\nabla \times \boldsymbol{F} = 0\) everywhere
2. \(\int_a^b \boldsymbol{F} \cdot \partial{\boldsymbol{l}}\) is path independent for any two points
3. \(\oint \boldsymbol{F} \cdot \partial{\boldsymbol{l}} = 0\) for any closed loop
4. \(\boldsymbol{F}\) is grad of some scalar: \(\boldsymbol{F} = -\nabla{V}\)

Conclusion: If curl of vector field \(\boldsymbol{F}\) vanishes everywhere, then \(\boldsymbol{F}\) can be expressed as a gradient of a scalar potential.
\(\nabla \times \boldsymbol{F} = 0 \Leftrightarrow \boldsymbol{F} = -\nabla{V}\)

Theorem 2: Divergence-less (solenoidal) fields

1. \(\nabla \cdot \boldsymbol{F} = 0\) everywhere
2. \(\int \boldsymbol{F} \cdot \partial{\boldsymbol{a}}\) is independent of surface for any given boundary line
3. \(\oint \boldsymbol{F} \cdot \partial{\boldsymbol{a}} = 0\) for any closed surface
4. \(\boldsymbol{F}\) is curl of some vector: \(\boldsymbol{F} = -\nabla{V}\)

Conclusion: If divergence of vector field \(\boldsymbol{F}\) vanishes everywhere, then \(\boldsymbol{F}\) can be expressed as a curl of a vector potential (\(\boldsymbol{A}\)).
\(\nabla \cdot \boldsymbol{F} = 0 \Leftrightarrow \boldsymbol{F} = \nabla \times \boldsymbol{A}\)

Vector potential is not unique - gradient of any scalar function can be added to \(\boldsymbol{A}\) without affecting the curl, since curl of the gradient is 0.
Any vector field can be written as gradient of a scalar plus curl of a vector
\(\boldsymbol{F} = \nabla{V} + \nabla \times \boldsymbol{A}\) (always).