Title: Theory / Important / Math / Analysis / Div, Grad, Curl and all that;

Name: An arrow in radial direction
Definition: An arrow which is aligned with the origin
Page: 4
Chapter: 1

Greater0: \(F(x,y ) = \mathbf{i}x+\mathbf{j}y\)
Page: 4
Chapter: 1
Smaller to greater: represents the function
Hide middle: x
Greater to smaller: is represented by

Greater0: \(F(x,y ) = \frac{-\mathbf{i}y+\mathbf{j}x}{\sqrt{x^2+y^2}}\)
Page: 4
Chapter: 1
Smaller to greater: represents the function
Hide middle: x
Greater to smaller: is represented by

Name: The effect of \(\mathbf r_1\) on \(\mathbf r_0\)
Definition: \(\varepsilon_0\) the permitivite of free space
Page: 6
Chapter: 1
Variables: \(\mathbf r_0\) and \(\mathbf r_1\) particles of charge \(q_0\) and \(q_1\)
Definition2: \(\mathbf {\hat u}\) the unit vector from \(r_1\) to \(r_0\)
Conjdef: is \(\frac{1}{4\pi\varepsilon_0}\frac{q_0q_1}{d^2}\mathbf {\hat u}\) with
Definition3: \(d\) the distance between \(r_0\) and \(r_1\)
Denoted by: \(\mathbf F\)

Name: The principle of superposition
Definition: If \(F1\) and \(F2\) is the action of \(q1\) and \(q2\) on \(q\), then the combined action of \(q1\) and \(q2\) on \(q\) is \(F1+F2\)
Page: 6
Chapter: 1

Name: The electrostactic field generated by \(\mathbf r\)
Definition: The function associating at each point \(p\) at distance \(d\) of \(\mathbf r\)
\(\frac{1}{4\pi\varepsilon_0}\frac{q}{d^2}\mathbf {\hat u}\) with \(\mathbf {\hat u}\) the unit vector from \(r_1\) to \(r_0\)
Page: 7
Chapter: 1
Variables: \(\mathbf r\) a particle of charge \(q\)
Context: electrostatic
Notation: \(\mathbf {E(r)}\)
Intuition: to get the effect of \(\mathbf r\) on a particle, you just have to multiply the charge of the particle with the value of the field at its position

Question: what mean "formal" in "formal solution"
Answer: Generally useless. Otherwise it would not be specified
Page: 11
Chapter: 2

Name: A vector normal to \(S\) at \(P\)
Definition: A vector orthogonal to all vectors tangeant to \(S\) at \(P\)
Page: 13
Chapter: 2
Variables: \(S\) a surface
\(P\in S\) a point
Example: the vector \(\mathbf k\)
Example2: radial vector, passing through the origin and \(P\)
Applied to: the xy-plane
Applied to2: \(\sqrt{1-x^2-y^2}\) (e.g. a half-sphere)

Page: 13
Implementation: Compute the cross product of two distinct tangeant vectors of \(S\) at \(P\)
Chapter: 2
Variables: \(S\) a surface
\(P\in S\) a point
Returns: A vector normal to \(S\) at \(P\)

Smaller0: \[\int\int_SG(x,y,z)\mathtt dS\]
Greater0: \[\int\int_{R}G(x,y,f(x,y))\sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}\mathtt dx\mathtt dy\]
Page: 20, 24
Chapter: 2
Variables: \(R\subseteq\mathbb R^2\)
\(f:\mathbb R^2\to\mathbb R\)
\(S\) the surface \(\{(x,y,f(x,y))\mid x,y\in R\}\)
\(G: S\to\mathbb R\)
Smaller to greater: =
Hide middle: y
Name: Surface integral
Intermediate: \[\lim_{\substack{N\to\infty\\\Delta S_i\to 0\\ \cup S_i=S}\sum_iG(x_i, y_i, z_i)\Delta S_i}\]

Smaller0: \(\mathbf{\hat n\cdot k}\)
Greater0: \(\frac1{\sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}}\)
Page: 24
Chapter: 2
Variables: \(S\) a surface
\(z=f(x,y)\) defining \(S\)
\(\mathbf{\hat n}(x, y, z)\) the normal of \(S\) at \((x,y,z)\)
\(\mathbf {i, j, k}\) the basis
Smaller to greater: =

Name: The flux of \(\mathbf F\) over the surface \(S\)
Definition: [$]\int\int_S\mathbf F(x,y,z)\cdot\mathbf{\hat n}\dd S[/$]
Page: 31
Chapter: 2
Variables: \(S\) a surface
\(z=f(x,y)\) defining \(S\)
\(\mathbf F:S\to\mathbb R^3\)
Notation: \(\Delta^n\)
Extra: with \(F=\rho v\) for \(\rho\) the density.
Intuition: If \(\mathbf F\) is the speed of a liquid, then the result is the quantity of liquid going through \(S\)
Cardinal: unique up to homeomophism

Name: The surface area of \(S\)
Definition: \(\int\int_S dS\)
Page: 20
Chapter: 2
Variables: \(S\) a surface in \(\mathbb R^3\)

Question: How to do a surface integral \(\int\int_SG(x,y,z)dS\) when the surface can't be expressed as \(S=\{(x, y, f(x,y))\mid (x, y)\in R\}\) for some \(R\subseteq \mathbb R^2\)
Answer: If it can be \(\{(x, f(x, z), z)\mid (x, z)\in R\}\) or \(\{(f(y, z), y, z)\mid (y, z)\in R\}\) just exchange \(z\) with \(x\) or \(y\) and proceed as usual
Page: 27
Chapter: 2
Variables: \(S\) a surface in \(\mathbb R^3\)
Answer2: Create a partition \(S=\bigcup_i S_i\) and compute
\(\sum_i\int\int_{S_i}G(x,y,z)dS_i\)

Name: del
Definition: \(\mathbf i\frac\partial{\partial x} + \mathbf j\frac\partial{\partial y}+ \mathbf k\frac\partial{\partial z}\)
Page: 44
Chapter: 2

Smaller0: \(\int\int_S\mathbf F\cdot\hat{\mathbf n}d S\)
Greater0: \(\int\int\int_V\nabla\cdot \mathbf F dV\)
Page: 45, 47
Chapter: 2
Variables: \(\mathbf F:\mathbb R^3\to\mathbb R^3\)
\(V\) a volume with boundary \(S\)
Smaller to greater: =
Name: The divergence theorem
Name2: Gauss theorem

Smaller0: \(\frac{\partial\rho}{\partial t}\)
Greater0: \(-\nabla\cdot(\rho v)\)
Page: 51
Chapter: 2
Variables: \(\rho(x, y, z, t)\) the density
\(v(x, y, z, t)\)the velocity
Smaller to greater: =
Name: A continuity equation
Context: Some fluid that is moving, but not created nor destructed

Name: The tangeant of \(C\) at \(r(s)\)
Definition: \(\mathbf i\frac{\Delta x}{\Delta s} + \mathbf j\frac{\Delta y}{\Delta s} + \mathbf k\frac{\Delta z}{\Delta s}\)
Page: 68
Chapter: 3
Variables: \(C\subseteq\mathbb R^3\) a curve parametrized by \(r(s) = (x(s), y(s), z(s))_{s\in[0, l]}\)
Notation: \(\hat{\mathbf t}\)

Name: An irrotational flow
Definition: A flow \(v\) such that \(\mathtt{div} v= 0\)
Page: 89
Chapter: 3

Name: A capping surface of the path \(C\)
Definition: Any surface whose boundary is \(C\)
Page: 93
Chapter: 3
Variables: \(C\subseteq\mathbb R^3\) a closed path

Smaller0: \(R\) is simply connected
Greater0: \(\int\int_S\hat{\mathbf n}\cdot\nabla\times \mathbf FdS\)
Page: 97
Chapter: 3
Variables: \(R\subseteq\mathbb R^3\) a region
\(S\subseteq R\) a surface
\(C\) the boundary of \(S\), a closed curve
\(\mathbf F:R\to\mathbb R^3\)
Assuming: \(\mathbf F\) differentiable
Name: Stoke's theorem
Greater1: \(\oint_C \mathbf{F\cdot\hat t}ds\)
Connect Greater0: =

Name: A simply connected region
Definition: A region such that any loop inside of it can be reduced to a point without leaving the region.
Page: 102
Chapter: 3

Smaller0: \(\oint_C\mathbf{F\cdot\hat t} = 0\) for all closed paths \(C\subseteq R\)
Greater0: \(\nabla\times\mathbf F = 0\)
Page: 103, https://en.wikipedia.org/wiki/Conservative_vector_field#Irrotational_vector_fields
Chapter: 3
Variables: \(R\subseteq\mathbb R^3\) a region
\(\mathbf F:R\to\mathbb R^3\)
Counterexample: [$$]\mathbf F=\defFun{\{(x, y, z)\in\mathbb R^3\mid z\ne 0\}}{\mathbb R^3}{(x, y, z)}{\left(-\frac{y}{x^2+y^2}, \frac{x}{x^2+y^2}, 0\right)}[/$$]
\(\nabla\times \mathbf F=0\) but \(\oint_C\mathbf F\cdot \mathbf{\hat t} dC = 2\pi\) for \(C\) the unit circe in the \(xy\)-plane

Smaller0: \(\oint_C\mathbf{F\cdot\hat t} = 0\) for all closed paths \(C\subseteq R\)
Greater0: \(\nabla\times\mathbf F = 0\)
Page: 103
Chapter: 3
Variables: \(R\subseteq\mathbb R^3\) a region
\(\mathbf F:R\to\mathbb R^3\)
Smaller to greater: iff
Assuming: \(R\) is simply connected
Intermediate: there exists \(\psi:R\to\mathbb R\) such that \(\mathbf F = \nabla\psi\)

Smaller0: \(\psi\) has a continuous second derivative
Greater0: \(\nabla\times(\nabla\psi)\)
Page: 119
Chapter: 3
Variables: \(R\subseteq\mathbb R^3\)
\(\psi:R\to\mathbb R\)
Greater1: 0
Greater2: \(\int_C\nabla\psi\cdot\mathbf tds\) is independant of path
Connect Greater1: =
Connect Greater2: and

Definition: \(\mathbf F\to\frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y}+ \frac{\partial F_z}{\partial z}\)
Page: 38, 41, 44
Chapter: 2
Notation: \(\nabla\cdot\)
Name: Divergence
Domain: \(\mathbb R^3\to\mathbb R^3\)
Codomain: \(\mathbb R\)
Abbreviation: \(\texttt{div}\)
Definition2: \(\lim_{\substack{\Delta V\to 0\\x\in V}}\frac{1}{\Delta V}\int\int_{S\in V}F dS\) with \(V\) a surface containing \(x\)
Intuition: Function that associates to a point in the space how much the field \(\mathbf F\) diverges from that point

Definition: \(F\mapsto\left|\begin{array}{lll}\mathbf i&\mathbf j&\mathbf k\\\frac{\partial}{\partial x}&\frac{\partial}{\partial y}&\frac{\partial}{\partial z}\\F_x&F_y&F_z\end{array}\right|\)
Page: 81
Chapter: 3
Notation: \(\nabla\)
Name: curl
Domain: \(\mathbb R^3\to\mathbb R^3\)
Definition2: \(\nabla\times\)
Intuition: When evaluated at \(r=(x, y, z)\) you get an operator that associates \(\hat{\mathbf n}\) to the rotation around \(r\)

Definition: \(\mathbf i\frac{\partial}{\partial x} + \mathbf j\frac{\partial}{\partial y} + \mathbf k\frac{\partial}{\partial j}\)
Page: 118
Chapter: 3
Name: Gradiant
Domain: \(\mathbb R^3\to\mathbb R\)
Codomain: \(\mathbb R^3\to\mathbb R^3\)
Abbreviation: grad
Definition2: \(\psi\mapsto\nabla\psi\)
Intuition: The direction in which the function increases the most, and how much it increases

Definition: \(\frac{\partial^2}{\partial x} + \frac{\partial^2}{\partial y}+ \frac{\partial^2}{\partial z}\)
Page: 122
Chapter: 3
Notation: \(\nabla^2\)
Name: Laplacian
Domain: \(\mathbb R^3\to\mathbb R\)
Codomain: \(\mathbb R^3\to\mathbb R^3\)
Definition2: \(\nabla\cdot\nabla\)

Smaller0: \(\frac d{ds}F(x, y, z)\)
Greater0: \(\hat{\mathbf u}\cdot\nabla F\)
Page: 122
Chapter: 3
Variables: \(C\subseteq\mathbb R^3\) a curve
\(F:\mathbb R^3\to\mathbb R\)
\(r=(x,y,z)\in C\)
\(\mathbf {\hat u}\) the direction of \(C\) at \(r\)
Smaller to greater: =