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Generality: Divergence and Directional derivative

 * We want to know the tangent of a function along a particular direction:
 * for an univariate function, there is only one direction;
 * for a multivariate function, there are many.
 * Divergence $$\nabla f$$ is the general formula for univariate and multivariate functions, it naturally points to the direction of maximum tangent (change rate).
 * The change rate (tangent) along a particular direction $$v$$ is: $$\nabla_v f = \nabla f \cdot v$$.
 * Thus, divergence is the generality of tangent under superposition principle; and multivariate function is the generality of univariate function under divergence - we don't treat it formally by means of mathematical abstraction, i.e., 1 is a special case of n.

Generality: Line integral (Integration along a line)

 * Topologically, line integral is the one-dimensional integral, the result is an area. It is the generalization of integration along a line (path, curve), with line the general term of one-dimensional manifold; the proper name is path integral or curve integral:
 * $$\int_C f ds$$
 *  C  denotes "curve",  ds  is the infinitesimal length of abscissa. The abscissa is the integration path (curve).


 * Line integral is applicable in n-dimensional space where $$n \geq 2$$, the principle is the same, though the function being integrated is ( n - 1 )-variate; the result is always an area in the n-dimensional space.
 * In 2D, $$f = f(x)$$, the abscissa is the  X-axis :
 * $$ds = dx$$.
 * The ordinate is  y , the path is in a scalar field  y = f(x) .
 * In 3D, $$f = f(x, y)$$, the abscissa is the path (curve) on  XY-plane :
 * $$ds = \sqrt{dx^2 + dy ^2} = \sqrt{1 + dy^2/dx^2}\ dx$$.
 * The ordinate is  z , the path (curve) is in a scalar field  z = f(x, y) .

Line integral on vector field

 * Line integral initially works on scalar field. In $$\int_C f ds$$, $$f = f(x, y)$$ is the scalar field.
 * When working on vector field, $$f = \vec{E} \cdot \vec{r}$$ is a dot product, $$\vec{E}$$ is a vector field, and $$\vec{r}$$ is a vector function.
 * Paul's Online > Line Integrals of Vector Fields

Coordinate system

 * A coordinate system is to use a tuple of coordinates to represent a point in the space.
 * Polar coordinate system uses $$(r, \theta)$$, the configuration is a circle. The derived coordinate systems:
 * Cylindrical coordinate system: configuration is cylinder, coordinates are $$(r, \theta, h)$$.
 * Spherical coordinate system: configuration is sphere, coordinates are $$(r, \theta, \phi)$$.
 * Cartesian coordinate system uses $$(x, y, z)$$, the configuration is a cuboid.

Points

 * Parametric coordinate form works for all coordinate systems:
 * $$(x(t),y(t),z(t))$$, $$(r(t),\theta(t))$$, $$(r(t),\theta(t),h(t))$$, $$(r(t),\theta(t),\phi(t))$$
 * These are just point sets, these points have no value attached to them.
 * Parametric coordinate form, function, and scalar field
 * Although in cartesian coordinates, we can say that $$f:(x(t),y(t)) \rightarrow z(t)$$ represents a function and thus a scalar field, but this is just a theoretical function, there is no formula we can use to manipulate the scalar field (such as obtaining its gradient). Thus, in general we don't take parametric coordiate form as the representation of scalar field nor function.

Scalar field

 * A scalar field is a scalar function that gives a scalar value to each point in the space:
 * $$f(P(\alpha, \beta, \gamma)) = \lambda, \lambda \in R$$
 * Practically, the function of a scalar field must have a formula, thus we can obtain its analytic values such as gradient. E.g.:
 * $$f(x,y,z) = 2x + 3y + 4z$$ is a scalar field in cartesian coordinates.
 * $$T(r, \theta, \phi) = \left[T_0 + T_1 sin^2\theta + T_2(1+sin\phi)\right]e^{-\alpha(r-R)}$$ is a scalar field on the Earth that represents temperature, see MIT - Fields.

Vector

 * A vector is a directional line-segment $$\overrightarrow{PQ} = (c-a, d-b), P = P(a,b), Q = Q(c,d)$$.
 * Which is a vector $$(c-a)\hat{i} + (d-b)\hat{j}$$ with $$P(a,b)$$ as the origin point, $$Q(c,d)$$ as the endpoint.
 * $$(a, b)$$ and $$a\hat{i} + b\hat{j}$$ represent the same thing. The benefit of the latter form is to make use of vector products by basis vectors:
 * $$\hat{i} \cdot \hat{i} = 1$$, $$\hat{i} \times \hat{j} = k$$, etc.

Parametric form

 * Parametric form is $$(x(t), y(t), z(t))$$ or equivalently $$x(t)\hat{i} + y(t)\hat{j} + z(t)\hat{k}$$.
 * Parametric form is to draw a graph (curve), as the coordinates are constrained by parameter  t , thus a graph (curve) is drawn. E.g.:
 * $$(cos(t), sin(t))$$ or $$cos(t)\hat{i} + sin(t)\hat{j}$$ is a circle.
 * Note that, the origin of a vector can be any point in the space, and the plus-sign in vector form means vector sum.

Vector field

 * While the parametric vector form:
 * $$(x(t), y(t), z(t))$$ or equivalently:
 * $$x(t)\hat{i} + y(t)\hat{j} + z(t)\hat{k}$$
 * can only draw a graph (curve), because the constraint is a single parameter  t , the multivariate vector form:
 * $$\vec{F}(x,y) = (P(x,y),Q(x,y))$$, or equivalently:
 * $$\vec{F}(x,y) = P(x,y)\hat{i} + Q(x,y)\hat{j}$$
 * can assign an arbitrary vector (directional segment) to each point in coordinates  (x, y) .
 * That's because  P(x, y)  and  Q(x, y)  are scalar functions, and we can design arbitraries of them. E.g.:
 * $$\vec{F}(x,y) = 2\hat{i}$$ is an constant vector field, at each point the vector is the same.
 * $$\vec{F}(x,y) = (2x)\hat{i}$$ is a gradually growing vector field along the x-axis.
 * $$\vec{F}(x,y) = (2x)\hat{i} + (y)\hat{i}$$ is another gradually growing vector field with y-axis component.
 * Of course we can have parametric form $$\vec{F}(t) =\vec{F}(x(t),y(t)) = t\hat{i} + 2t\hat{i}$$, but this is a special case.
 * Ref: Paul's Online > Vector Fields

Dot-product and line-integral of vector and vector field

 * Dot product of two vector forms $$\vec{A}\cdot\vec{B}$$, if $$\vec{A}$$ is a force field, and $$\vec{B}$$ is the resultant path of a moving particle in the field, then the dot-product is the work per unit of physical magnitude (such as mass or electric charge). E.g.:
 * $$\vec{E}\cdot\vec{r}$$ is the work of an electric charge moving in an electric field, in which $$\vec{r}$$ is the resultant path.
 * In general, we do line integral to obtain the total work:
 * $$\int_C{\vec{E}\cdot d\vec{r}} = \int_C{\vec{E}(\vec{r}(t))\cdot d\vec{r}(t)} = \int_C{\vec{E}(\vec{r}(t))\cdot\frac{d\vec{r}(t)}{dt}dt} = \int_C{\vec{E}(\vec{r}(t))\cdot(\vec{r}(t))'dt}$$


 * Thus, line integral is the integration of dot-product. That's why line integral is path independent.
 * Paul's Online > Line Integrals of Vector Fields

Potential function, Gradient field, Conservative (Path-independent), Fundamental theorem of line integral (Gradient theorem)

 * Force field is the gradient field of a potential field.
 * A potential field is a scalar field. A potential function is a scalar function that describes a potential field.
 * In a potential field, energy is stored at each point, called potential. The potential difference causes acceleration, thus the force field.
 * The gradient of a potential function describes a vector field called gradient field, which is the force field. The gradient theorem is in fact the gradient theorem of a potential field.
 * Let $$\vec{F}$$ be the vector function of the force field, $$f$$ the potential function, thus $$\nabla f = \vec{F}$$:
 * $$\int_C\vec{F}\cdot d\vec{r} = \int_C\nabla f\cdot d\vec{r}$$


 * The line integral of the gradient field of a potential function is path-independent for two reasons:
 * It is an integration of dot-product, i.e., projection. The projection is the same regardless of the curve being projected to the same segment, if the space is uniform.
 * By the theory of potential, if the path is closed, there is no resultant potential difference, and thus no energy loss. This is called conservative. The gradient field (of a potential function) is thus called conservative vector field.
 * Scalar field > Uses in physics
 * [MathWorld > Gradient Theorem]

Line integral and Exact differential

 * $$\int_C{\vec{E}\cdot d\vec{r}} = \int_C(M(x,y)\hat{i}+N(x,y)\hat{i})\cdot(dx\hat{i}+dy\hat{j}) = \int_C Mdx + Ndy$$


 * $$Mdx + Ndy$$ is called a differential form.
 * $$\int Mdx + Ndy$$ is commonly understood as line integral because $$\int Mdx + Ndy = \int\langle M, N\rangle\cdot \langle dx, dy\rangle$$.
 * If $$Mdx + Ndy = f_x dx + f_y dy = df$$, it is called a total differential.


 * MIT OCW > V2.2-3 Gradient Fields and Exact Differentials

Polar coordinate system
Formulae of geometries:
 * To represent a set of points, use parametric form or vector form, they are equivalent:
 * Polar: $$(r(t), \theta(t))$$ or $$r(t)\hat{r} + \theta(t)\hat{\theta}$$
 * Cartesian: $$(x(t), y(t), z(t))$$ or $$x(t)\hat{i} + y(t)\hat{j} + z(t)\hat{k}$$