To be Studied (but outside of scope…)

• Curvilinear coordinates

https://en.wikipedia.org/wiki/Curvilinear_coordinates

• Tensor

https://en.wikipedia.org/wiki/Tensor

Vectors, Operators and Derivatives:

Different types of functions

Multivariable Chain Rule

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Let $\phi = f(x,y)$, $x=g(t)$ and $y=h(t)$, then

$$ \frac{d\phi}{dt}=\frac{d f}{dt}=\frac{\partial f}{\partial x}\frac{d x}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt} $$

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Directional Derivative

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Def.

Let a unit vector be $\hat v =(\alpha ,\beta)$ and $\phi = f(x,y)$.

The directional derivative at direction $\hat v$ is defined as

$$ D_{\hat v}f=\alpha \frac{\partial f}{\partial x}+\beta \frac{\partial f}{\partial y}= (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y})\cdot \hat v $$

• The idea can be expanded to 3D function.
• The directional derivative measurems the slope of $f$ in direction of $\hat v$.
• The directional derivative is maximised in the direction of the gradient of the function, so that it satisfies $|D_{\hat v}f| \le |\nabla \cdot f|$.

Properties of directional derivatives:

• $D_{\hat v}(\lambda f)=\lambda \cdot D_{\hat v}f$
• $D_{\hat v}(f+g)=D_{\hat v}f+D_{\hat v}g$
• $D_{\hat v}f\cdot g =(D_{\hat v}f)g+(D_{\hat v}g)f$ </aside>

Gradient

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Def. Nabla operator and Gradient operator

For a scalar field $f: \mathbb{R}^3 \rightarrow \mathbb{R}$, define the following operators:

Nabla operator:

$$ \nabla = \frac{\partial }{\partial x}\vec i + \frac{\partial }{\partial y} \vec j+ \frac{\partial }{\partial z}\vec k $$

When nabla operator operates on $f$, a scalar field, it becomes gradient operator:

$$ \text{grad}\ f=\nabla \cdot f = \frac{\partial f}{\partial x}\vec i + \frac{\partial f}{\partial y} \vec j+ \frac{\partial f}{\partial z}\vec k $$

Physically, $\nabla \cdot f$ is a vector perpendicular to the surface $f(x,y,z)= \text{constant}$.

• $\nabla \cdot f$ points toward the direction of steepest ascent of the function $f$; similarly, $-\nabla \cdot f$ points toward the direction of steepest descent of the function $f$.
• By letting $D_{\hat v}f=\alpha f_x+\beta f_y=0$, it follows that $(-\frac{\partial f}{\partial y}, \frac{\partial f}{\partial x})^T$ and $(\frac{\partial f}{\partial y}- \frac{\partial f}{\partial x})^T$ point toward the direction of least change of the function $f$. </aside>

Divergence

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Def. divergence (in Cartesian coordinates)

Divergence of a vector field $\vec F = F_x \vec i + F_y \vec j + F_z \vec k$ is defined as the scalar function:

$$ \text{div}\ \vec F = \nabla \cdot \vec F = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} $$

A vector field with $\text{div}\ \vec F =0$ everywhere is called solenoidal. Any closed surface has no net flux across it in solenoidal vector field. Constant vector field is always solenoidal.

Physically, the divergence of a vector field is the extent to which the flux of vector field behaves like a source or a sink at a given point. The greater the flux of field through a small surface enclosing a given point, the greater the value of divergence at that point.

The divergence of a certain point in vector field can be considered as the outward flux of the infinitesimally-sized box at that point.

$\text{div}\ \vec F >0$ implies net outflow, $\text{div}\ \vec F <0$ implies net inflow.

Properties:

• Linear combination: $\nabla \cdot (a\vec F+b\vec G)= a \ \nabla \cdot \vec F + b \ \nabla \cdot \vec G$
• Product rule 1: $\nabla \cdot (\psi \vec F)= (\nabla \cdot \psi)\vec F + \psi (\nabla \cdot \vec F)$, where $\psi$ is a scalar function. </aside>

Total Differential

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Def.

Total differential $d\phi$ of a scalar field $\phi (x,y,z)$ is defined as

$$ d\phi=\frac{\partial \phi}{\partial x}dx+\frac{\partial \phi}{\partial y}dy+\frac{\partial \phi}{\partial z}dz $$

Total differential tells the change of $\phi$ as one moves infinitesimal amounts $dx$, $dy$ and $dz$ in the (x,y,z) directions.

Similarly, total differential $d\vec F$ of a vector field $\vec F (x,y,z)$ is defined as

$$ d\vec F=\frac{\partial \vec F}{\partial x}dx+\frac{\partial \vec F}{\partial y}dy+\frac{\partial \vec F}{\partial z}dz $$

Geometric intepretation:

For a 2D curve which can be described by $\vec r = f(x,y$), $x=x(t)$, $y=y(t)$,

let $s$ be the arc length, the unit tangent vector is

$$ \hat \tau=\frac{d\vec r}{ds} $$

where $d\vec r$ can be obtained by total differential.

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Multiple Integrals

Double Integral Definition