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Vector Fields

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A vector field F(x,y,z)\mathbf{F}(x,y,z) on R3\mathbb{R}^3
Virtually all of physics deals with fields. The electric and magnetic fields of electromagnetism, the gravitational and other force fields of mechanics, the velocity fields of fluid flow, and the wave function of quantum mechanics are all “fields”. Fields are also used in epidemiology and population studies.
VECTOR FIELD
A vector field on Rn\mathbb{R}^n is a function whose input is a point in Rn\mathbb{R}^n and whose output is a vector in Rn\mathbb{R}^n emanating from that point.
By “field” we mean data that varies from point to point. Some fields, like temperature or pressure distribution, are scalar fields: they associate a number to every point. Some fields, like the Newtonian gravitation field, are best modeled by vector fields, which associate a vector to every point.
The identity vector field F(x,y)=(x,y)\mathbf{F}(x,y)=(x,y)
Let start with simplest example of the vector field in R2\mathbb{R}^2
F(x,y)=(x,y).\begin{equation*}\mathbf{F}(x,y)=(x,y).\end{equation*}
At every nonzero point, the vector points directly away from the origin and has magnitude x2+y2\sqrt{x^2+y^2}.
Newton's law of gravitation states that the magnitude of the force between masses mm and MM, separated by a distance rr, is
∥F∥=mMGr2,\begin{equation*}\lVert\mathbf{F}\rVert=\frac{mMG}{r^2},\end{equation*}
where GG is the gravitational constant. Place the mass MM at the origin and the mass mm at r=(x,y,z)\mathbf{r}=(x,y,z). Since r=∥r∥r=\lVert\mathbf{r}\rVert and the unit vector pointing toward the origin is −r/∥r∥-\mathbf{r}/\lVert\mathbf{r}\rVert, the force field is
F(r)=−mMG∥r∥3r.\begin{equation*}\mathbf{F}(\mathbf{r}) = -\frac{mMG}{\lVert\mathbf{r}\rVert^3}\mathbf{r}.\end{equation*}
The gravitational vector field
This is a compact form of the gravitational field. Since r=(x,y,z)T\mathbf{r}=(x,y,z)^{\mathsf{T}} and
∥r∥=x2+y2+z2,\begin{equation*}\lVert\mathbf{r}\rVert=\sqrt{x^2+y^2+z^2},\end{equation*}
the field can also be written in terms of its component functions as
F(x,y,z)=(−mMGx(x2+y2+z2)3/2−mMGy(x2+y2+z2)3/2−mMGz(x2+y2+z2)3/2).\begin{equation*}\mathbf{F}(x,y,z)=\begin{pmatrix}-\dfrac{mMGx}{(x^2+y^2+z^2)^{3/2}} \\-\dfrac{mMGy}{(x^2+y^2+z^2)^{3/2}} \\-\dfrac{mMGz}{(x^2+y^2+z^2)^{3/2}}\end{pmatrix}.\end{equation*}
In this section we define an integral that is similar to a single integral except that instead of integrating over an interval [a,b][a, b] we integrate over a curve C C . Such integrals are called line integrals, although “curve integrals” would be better terminology. They were invented in the early 19th century to solve problems involving fluid flow, forces, electricity, and magnetism.
A partition of the parameter interval induces points and subarcs on the plane curve CC; the blue point corresponds to the sample parameter ti∗t_i^*.
Let CC be a smooth plane curve parametrized by
r(t)=(x(t),y(t)),a≤t≤b,\begin{equation*}\mathbf{r}(t)=\bigl(x(t),y(t)\bigr), \qquad a\leq t\leq b,\end{equation*}
where r′\mathbf{r}' is continuous and nonzero. Choose a partition
a=t0<t1<⋯<tn=b\begin{equation*}a=t_0<t_1<\cdots<t_n=b\end{equation*}
and a sample point ti∗∈[ti−1,ti]t_i^*\in[t_{i-1},t_i] in each subinterval. Let Δsi\Delta s_i denote the length of the corresponding subarc of CC.
SCALAR LINE INTEGRAL IN THE PLANE
If ff is defined on a smooth plane curve CC, its line integral along CC with respect to arc length is
∫Cf ds=lim⁡max⁡iΔti→0∑i=1nf(x(ti∗),y(ti∗)) Δsi,\begin{equation*}\int_C f\,\mathrm{d}s=\lim_{\max_i\Delta t_i\to0}\sum_{i=1}^nf\bigl(x(t_i^*),y(t_i^*)\bigr)\,\Delta s_i,\end{equation*}
provided that the limit exists, where Δti=ti−ti−1\Delta t_i=t_i-t_{i-1}.
The arc-length differential satisfies
ds=∥r′(t)∥ dt=(dxdt)2+(dydt)2 dt.\begin{equation*}\mathrm{d}s=\lVert\mathbf{r}'(t)\rVert\,\mathrm{d}t=\sqrt{\left(\frac{\mathrm{d}x}{\mathrm{d}t}\right)^2+\left(\frac{\mathrm{d}y}{\mathrm{d}t}\right)^2}\,\mathrm{d}t.\end{equation*}
Therefore, if ff is continuous on CC, the line integral can be evaluated from any smooth parametrization by
∫Cf(x,y) ds=∫abf(x(t),y(t))(dxdt)2+(dydt)2 dt.\begin{equation*}\int_C f(x,y)\,\mathrm{d}s=\int_a^bf\bigl(x(t),y(t)\bigr)\sqrt{\left(\frac{\mathrm{d}x}{\mathrm{d}t}\right)^2+\left(\frac{\mathrm{d}y}{\mathrm{d}t}\right)^2}\,\mathrm{d}t.\end{equation*}
Let CC be a smooth space curve parametrized by
r(t)=(x(t),y(t),z(t)),a≤t≤b.\begin{equation*}\mathbf{r}(t)=\bigl(x(t),y(t),z(t)\bigr),\qquad a\leq t\leq b.\end{equation*}
As in the plane, divide CC into subarcs of lengths Δsi\Delta s_i and choose a sample point ti∗t_i^* on each corresponding parameter interval.
SCALAR LINE INTEGRAL IN SPACE
If ff is continuous on a smooth space curve CC, then its line integral along CC with respect to arc length is
∫Cf(x,y,z) ds=lim⁡max⁡iΔti→0∑i=1nf(x(ti∗),y(ti∗),z(ti∗)) Δsi.\begin{equation*}\int_C f(x,y,z)\,\mathrm{d}s=\lim_{\max_i\Delta t_i\to0}\sum_{i=1}^nf\bigl(x(t_i^*),y(t_i^*),z(t_i^*)\bigr)\,\Delta s_i.\end{equation*}
Since
ds=∥r′(t)∥ dt=(dxdt)2+(dydt)2+(dzdt)2 dt,\begin{equation*}\mathrm{d}s=\lVert\mathbf{r}'(t)\rVert\,\mathrm{d}t=\sqrt{\left(\frac{\mathrm{d}x}{\mathrm{d}t}\right)^2+\left(\frac{\mathrm{d}y}{\mathrm{d}t}\right)^2+\left(\frac{\mathrm{d}z}{\mathrm{d}t}\right)^2}\,\mathrm{d}t,\end{equation*}
the integral can be evaluated as
∫Cf(x,y,z) ds=∫abf(x(t),y(t),z(t))(dxdt)2+(dydt)2+(dzdt)2 dt=∫abf(r(t))∥r′(t)∥ dt.\begin{align*}\int_C f(x,y,z)\,\mathrm{d}s&=\int_a^bf\bigl(x(t),y(t),z(t)\bigr)\sqrt{\left(\frac{\mathrm{d}x}{\mathrm{d}t}\right)^2+\left(\frac{\mathrm{d}y}{\mathrm{d}t}\right)^2+\left(\frac{\mathrm{d}z}{\mathrm{d}t}\right)^2}\,\mathrm{d}t \\&=\int_a^b f\bigl(\mathbf{r}(t)\bigr)\lVert\mathbf{r}'(t)\rVert\,\mathrm{d}t.\end{align*}
The one-dimensional Fundamental Theorem of Calculus states that, for a continuously differentiable function hh,
∫abh′(t) dt=h(b)−h(a).\begin{equation*}\int_a^b h'(t)\,\mathrm{d}t=h(b)-h(a).\end{equation*}
For a gradient field, the corresponding line integral also depends only on the endpoints of the curve.
FUNDAMENTAL THEOREM FOR LINE INTEGRALS
Let U⊆RnU\subseteq\mathbb{R}^n be open, let f∈C1(U)f\in C^1(U), and let C⊂UC\subset U be a smooth curve parametrized by r:[a,b]→U\mathbf{r}:[a,b]\to U. Then
∫C∇f⋅dr=f(r(b))−f(r(a)).\begin{equation*}\int_C \nabla f\mathbin{\boldsymbol{\cdot}}\mathrm{d}\mathbf{r}=f\bigl(\mathbf{r}(b)\bigr)-f\bigl(\mathbf{r}(a)\bigr).\end{equation*}
Write r(t)=(x(t),y(t),z(t))\mathbf{r}(t)=(x(t),y(t),z(t)); the same argument applies in any dimension. By the definition of a line integral and the multivariable chain rule,
∫C∇f⋅dr=∫ab∇f(r(t))⋅r′(t) dt=∫ab(∂f∂xx′(t)+∂f∂yy′(t)+∂f∂zz′(t))dt=∫abddtf(r(t)) dt=f(r(b))−f(r(a)),\begin{align*}\int_C \nabla f\mathbin{\boldsymbol{\cdot}}\mathrm{d}\mathbf{r}&=\int_a^b \nabla f\bigl(\mathbf{r}(t)\bigr)\mathbin{\boldsymbol{\cdot}}\mathbf{r}'(t)\,\mathrm{d}t \\&=\int_a^b\left(\frac{\partial f}{\partial x}x'(t)+\frac{\partial f}{\partial y}y'(t)+\frac{\partial f}{\partial z}z'(t)\right)\mathrm{d}t \\&=\int_a^b \frac{\mathrm{d}}{\mathrm{d}t}f\bigl(\mathbf{r}(t)\bigr)\,\mathrm{d}t=f\bigl(\mathbf{r}(b)\bigr)-f\bigl(\mathbf{r}(a)\bigr),\end{align*}
where the final equality follows from the one-dimensional Fundamental Theorem of Calculus.
The theorem also holds for piecewise-smooth curves: subdivide the curve into finitely many smooth pieces, apply the theorem to each piece, and sum the resulting integrals. The contributions at adjacent endpoints cancel.
In the plane, a vector field is usually written as
F(x,y)=P(x,y)i+Q(x,y)j.\begin{equation*}\mathbf{F}(x,y)=P(x,y)\mathbf{i}+Q(x,y)\mathbf{j}.\end{equation*}
The arrow at (x,y)(x,y) describes the direction and magnitude of the field at that point.
Vector fields appear in many parts of mathematics and physics. A fluid velocity field tells us how a fluid parcel moves, a gravitational field tells us the force on a mass, and electric or magnetic fields describe forces on charged particles.
The divergence of a two-dimensional vector field F=Pi+Qj\mathbf{F}=P\mathbf{i}+Q\mathbf{j} is
div⁡F=∂P∂x+∂Q∂y.\begin{equation*}\operatorname{div}\mathbf{F}=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}.\end{equation*}
It measures the local tendency of the field to flow outward from a point. Positive divergence means the point behaves like a source, negative divergence means it behaves like a sink, and zero divergence means there is no local creation or destruction of flow.
For incompressible fluid flow, div⁡F=0\operatorname{div}\mathbf{F}=0 everywhere: locally, fluid volume is preserved.
For an incompressible fluid, the mathematical condition is
div⁡F=0.\begin{equation*}\operatorname{div}\mathbf{F}=0.\end{equation*}
The flow can bend, accelerate, and move around obstacles, but it does not appear or disappear inside the region.
The curl of a two-dimensional vector field F=Pi+Qj\mathbf{F}=P\mathbf{i}+Q\mathbf{j} is the scalar
curl⁡F=∂Q∂x−∂P∂y.\begin{equation*}\operatorname{curl}\mathbf{F}=\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}.\end{equation*}
It measures the local tendency of the field to rotate around a point.
Positive curl corresponds to counterclockwise local rotation, while negative curl corresponds to clockwise local rotation.
The value of curl changes from point to point in a vector field.
Because curl is computed from partial derivatives, it is determined by how the field changes near a point, not by the whole picture at once. Moving the sample point through the same field can therefore produce positive, negative, or nearly zero curl.

References

  • [HubbardHubbard2015]Hubbard, John H., and Barbara Burke Hubbard. Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. Fifth edition. Matrix Editions, 2015.
  • [StewartEtAl2021]Stewart, James, Daniel K. Clegg, and Saleem Watson. Calculus: Early Transcendentals, Metric Edition. Ninth edition. Cengage, 2021.