Concepts

A Path in Rn\mathbb{R}^n is a continuous function γ:I↦Rn\gamma: I\mapsto \mathbb{R}^n where II is an interval in R\mathbb{R}. The image set γ(I)\gamma(I) is called a curve.

If I=[a,b]I=[a,b] is a closed and bounded interval, then γ(a),γ(b)\gamma(a), \gamma(b) are called the endpoints of γ\gamma.

Velocity Vector

Let γ:I↦R\gamma: I\mapsto\mathbb{R} be a differentiable path. Then

v(t)=γ′(t)\mathbf{v}(t) = \gamma'(t)

is called the velocity vector of the path. And its length ∥v(t)∥\|\mathbf{v}(t)\| is called the speed of the path.

Length of Path

Let γ:[a,b]↦Rn\gamma:[a,b]\mapsto \mathbb{R}^n be a differentiable path of class C1C^1, where a,b∈Ra,b\in\mathbb{R}. Then the length of γ\gamma is given by

∫ab∥γ′(t)∥dt\int_a^b \|\gamma'(t)\| dt

If a path can be partitioned into a finite number of C1C^1 paths, then we can its length by summing up the length of sub-paths.

Proof. We can break down the path γ\gamma into “many” segments. For each segment, suppose a small change Δt\Delta t, since the curve is differentiable, the length of this segment can be approximated by (Δx)2+(Δy)2+(Δz)2\sqrt{(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2}, where Δx≈x′(t)Δt\Delta x\approx x'(t) \Delta t and similarly for Δy,Δz\Delta y, \Delta z, thus this square root can be approximated by (x′)2+(y′)2+(z′)2Δt=∥γ′(t)∥Δt\sqrt{(x')^2 + (y')^2 + (z')^2}\Delta t=\|\gamma'(t)\|\Delta t. By summing over segments and taking limit, this is ∫∥γ′∥dt\int \|\gamma'\| \mathrm{d}t. ■\blacksquare

Differential Operators

  • a vector field on Rn\mathbb{R}^n is a function F:X↦Rn\mathbf{F}: X\mapsto\mathbb{R}^n where X⊂RnX\subset \mathbb{R}^n
  • a scalar field … f:X↦Rf:X\mapsto\mathbb{R}
  • gradient field or conservative vector field is a vector field F:X↦Rn,X⊂Rn\mathbf{F}:X\mapsto \mathbb{R}^n, X\subset\mathbb{R}^n, which is the gradient of some function f:X↦R,X⊂Rnf:X\mapsto \mathbb{R},X\subset\mathbb{R}^n

Del Operator

The del operator in Rn\mathbb{R}^n is defined by

∇=∂∂x1e1+∂∂x2e2+⋯+∂∂xnen\nabla = \frac{\partial}{\partial x_1}\mathbf{e}_1+\frac{\partial}{\partial x_2}\mathbf{e}_2+\dots+\frac{\partial}{\partial x_n}\mathbf{e}_n

It maps a scalar field to a vector field.

Proposition 1

If f:X↦Rf:X\mapsto\mathbb{R} be a scalar field of class C2C^2 where X⊂R3X\subset \mathbb{R}^3, then

∇×(∇f)=0\nabla\times(\nabla f)=0

Proof. ∇f=(fx,fy,fz)\nabla f=(f_x,f_y,f_z), then

∇×∇f=∣ijk∇x∇y∇zfxfyfx∣=(fzy−fyz)i−(fzx−fxz)j+(fyx−fxy)k=0\begin{aligned}\nabla\times\nabla f &=\begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\\nabla_x & \nabla_y & \nabla_z \\f_x & f_y & f_x\end{vmatrix}\\&= (f_{zy} - f_{yz})\mathbf{i} - (f_{zx}-f_{xz})\mathbf{j} + (f_{yx}-f_{xy})\mathbf{k} \\&= 0\end{aligned}

Proposotion 2

∇⋅(∇×F)=0\nabla\cdot(\nabla \times F)=0

Divergence 散度

divF=∇⋅F=∑i=1n∂Fi∂xi\text{div} F=\nabla\cdot F=\sum_{i=1}^n \frac{\partial F_i}{\partial x_i}

divF\text{div} F measures the net flow of the vector field.

For a point in the field, if we draw a region containing this point, then some vectors will go from outside to inside and some vectors from inside to outside. “Divergence” measures this difference between in-flow and out-flow: if there’s more in-flow than out-flow, the divergence is negative; if there’s more out-flow than in-flow, the divergence is positive.

If we shrink the region and take limit to make region approach the point, then the difference between in- and out-flow has a limit, which is the divergence at this point.

Curl 旋度

Let F:X↦R3F:X\mapsto \mathbb{R}^3

curlF=∇×F\text{curl} F=\nabla\times F

curlF\text{curl} F measures the twisting and circulation of the vector field. The norm of curl tells the strength of twisting.