Math Refresher

Vector Field $$\vec{F}(x, y, z) = \langle P(x, y, z),, Q(x, y, z),, R(x, y, z) \rangle$$

• for each point in 3D, a 3D vector exists
• for fluid flow, at any location in the fluid, the fluid is moving in a direction with a certain velocity
• for an electric field, at any point, the electric field points in some direction with a certain strength

Jacobian

$$\mathbf{J} = \begin{bmatrix} \frac{\partial P}{\partial x} & \frac{\partial P}{\partial y} \ \frac{\partial Q}{\partial x} & \frac{\partial Q}{\partial y} \end{bmatrix}$$

• intuitively, keeps track of the “forces” that act on a square at a location
• diagonal entries $\frac{\partial P}{\partial x}$ and $\frac{\partial Q}{\partial y}$ are stretching/squishing horizontally and vertically
• off-diagonal $\frac{\partial P}{\partial y}$ and $\frac{\partial Q}{\partial x}$ are sheer and rotation; square rotates or turns to parallelogram

Divergence $$\nabla \cdot \vec{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y}$$

• divergence gives local volumetric growth rate
• intuitively, how much a region “generates” a fluid, so sources have positive divergence, and sinks have negative divergence
• sum of the diagonal entries, so the sum of all stretching and squishing effects

Curl $$\nabla \times \vec{F} = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}$$

• captures rotation, “were a twig placed in the vector field, how would it rotate?”

Flux $$\Phi = \iint_S \vec{F} \cdot \hat{n} , \mathrm{d}A$$

• measures how much of a vector field passes perpendicularly through a surface
• $\vec{F}$ is the vector field at a point in space
• $ \hat{n}$ is the unit vector perpendicular to the surface
• $\mathrm{d}A$ is some infinitely small patch on the surface
• $\iint_S$ sums over the surface
• the dot product calculates the normal component of the vector field, perpendicular to the surface
• flux is a scalar number equal to total field passing through the surface

Divergence Theorem: Relationship between Divergence and Flux

• flux is defined over an area, while divergence applies to individual points $$\iint_S \vec{F} \cdot d\vec{A} = \iiint_V (\nabla \cdot \vec{F}) , dV$$
• the flux across a surface equals the divergence of the enclosed volume $$\nabla \cdot \vec{F} = \lim_{V \to 0} \frac{1}{V} \oint_S \vec{F} \cdot d\vec{A}$$
• divergence at a point is outward flux per unit volume; stronger divergence means bigger source

Integral Form of Gauss’s Law for Electric Fields

$$\oint_S \vec{E} \circ \hat{n} , da = q_{\text{enc}} / \varepsilon_0$$

Units

• Work is how energy goes in or out of an object; you work on an object when you apply a force over a displacement; $W = F \cdot d$
• Energy is the capacity to do work; $\Delta E = F \cdot d$ $$v^2 = v_0^2 + 2ad \implies ad = \frac{v^2 - v_0^2}{2}$$ $$m \cdot a \cdot d = \frac{1}{2}mv^2 - \frac{1}{2}mv_0^2$$ $$F \cdot d = \frac{1}{2}mv^2 - \frac{1}{2}mv_0^2$$ $$W = F \cdot d = \Delta E_k$$
• Volt is electric potential. “Mass” in “gravitational potential” is like “charge” in “electric potential”. Units of gravitational energy are $\text{J/kg}$ and units of volts are $\text{J/C}$ where $\text{C}$ is Coulombs. Textbook definition is potential difference between two points that imparts one joule of energy per coulomb of charge.
• potential $V$ vs potential energy $U$; potential is like an “object amount” standardized unit of potential energy. so electric potential is energy per charge $\text{J/C}$.
• electric field determines how much force will act per charge in the field. $\vec{E} = \frac{\vec{F}}{q}$; potential is energy per charge $\text{J/C}$. field is force per charge $\text{N/C}$. reminder: energy is force over a distance.
• force of the electric field is caused by a difference in electric potential between two points in space over a distance; the size of the potential difference over a distance determines the amount of force. So $E = -\frac{\Delta V}{\Delta x}$; units are $\text{V/m}$ or $\text{N/C}$
• flux is total electric field passing over a surface, so $\Phi_E = E \times A = \left(\frac{\text{V}}{\text{m}}\right) \times \text{m}^2 = \text{V} \cdot \text{m}$

Electric Flux of Closed Surface $$\Phi = \int_S \left( \vec{E} \cdot \hat{n} \right) \mathrm{d}a$$

• electric field strength at a point is proportional to density of field lines in a plane perpendicular to the field at the point
• integrating that field line density gives net number of field lines penetrating the surface
• electric flux is net number of field lines penetrating the surface
• say there is a closed 3D surface, like a box or circle; if electric field lines originate in the box and go outside the box, flux is positive
• if electric field lines originate outside the box and terminate inside the box, flux is negative
• if electric field lines originate and terminate outside the box, flux is zero
• electric field lines originate at a positive charge and terminate at a negative charge; arrow on line points direction positive charge would be pushed
• if positive charge in box, field lines originate in box and terminate outside, flux is positive
• if negative charge in box, flux is negative; if neutral charge in box, field is zero
• gauss’s law for electric fields = electric flux through closed surface is proportional to charge contained within the surface

Enclosed Charge $$q_{\text{enc}}$$

• any charge not enclosed will create field lines that pass through enclosed surface causing their flux contribution to be zero
• in real world, counting charge is hard; instead use charge density (over length, area, or volume) and multiply by length/area/volume; can take integral if not uniformly distributed

Permittivity $$\varepsilon_0$$

• in the fluid flowing analogy, field is direction and velocity of flowing water particles at every 3D point, electric flux is volume of water that flows through a cross-section of the pipe every second; positive charge is pump, negative charge is drain
• permittivity is resistance to water flow, and relates pump and volume of water flowing in pipe; volume = pump / permittivity; high permittivity means more resistance
• a conductor in an electric field will have its electrons leave the material causing a current; in an insulator, each atom will polarize and have their dipoles line up end-to-end
• negative charges will build up on the material’s surface close to the source of the electric field; positive charges will build up on the surface close to the sink of the electric field
• this charge buildup will create an opposing electric field that provides the “resistance”; the strength of this resistance determines the value of the permittivity
• permittivity’s units are $\mathbf{\text{C} / (\text{V} \cdot \text{m})}$; how much charge builds up for an applied electric field
• an interesting question is why is the permittivity of a vacuum is not zero if there are no atoms to create dipoles
  • quantum mechanics believes it comes from “virtual” electrons and positive charges, that pop in an out of existance but are unobservable, and create temporary dipoles
  • classical mechanics believes it to be a fundamental constant; a scaling factor of reality

Differential Form of Gauss’s Law for Electric Fields

$$\nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0}$$

• electric field diverges from positive charge and converges on negative charge

Positive Point Charge

• take two concentric spheres arount the point charge
• the flux of the surfaces is the same because all field lines entering the space between the spheres also leave
• although field strength drops by $1/r^2$, surface area increases by $r^2$, so the flux over the surface stays constant
• divergence at a point is outward flux per unit volume; since flux is constant, divergence is zero (everywhere except at point charge/origin)
• at origin, at an infinitely small point, outward flux is positive, because of the positive charge, so divergence is positive