[b][size=150]Vector fields in two dimension[/size][/b][br][br]Let [math]f(x,y)[/math] and [math]g(x,y)[/math] be functions defined on a domain [math]R[/math] in [math]\mathbb{R}^2[/math]. A [b]vector field[/b] on [math]R[/math] is a vector-valued function [math]\vec{F}:R\to \mathbb{R}^2[/math] such that for any [math](x,y)[/math] in [math]R[/math],[br][br][math]\vec{F}(x,y)=\langle f(x,y), g(x,y) \rangle = f(x,y)\vec{i}+ g(x,y)\vec{j}[/math][br][br]A vector field [math]\vec{F}=\langle f,g \rangle[/math] is continuous (or differentiable) on [math]R[/math] if [math]f[/math] and [math]g[/math] are continuous (or differentiable) on [math]R[/math].[br][br]The applet below illustrates a vector field [math]\vec{F}[/math] in 2D space - a vector (arrow) is assigned to every point in the rectangular domain [math][a,b]\times [a,b][/math]. [br][br]You can try the following examples of vector fields:[br][br][list=1][*][math]\vec{F}(x,y)=\langle 0, x\rangle[/math] (a shear field)[/*][*][math]\vec{F}(x,y)=\langle 16-y^2,0\rangle[/math] for [math]|y|\leq 4[/math] (a channel flow)[/*][*][math]\vec{F}(x,y)=\langle -y,x\rangle[/math] (a rotation field)[/*][*][math]\vec{F}(x,y)=s(x,y)\langle x,y \rangle[/math], where [math]s(x,y)[/math] is a real-valued function. It is called a [b]radial vector field[/b]. Of specific interest are the radial vector fields [math]\vec{F}(x,y)=\frac 1{|\langle x,y \rangle|^p}\langle x,y \rangle[/math], where [math]p[/math] is a real number.[/*][/list][br]

[b][size=150]Vector fields in three dimension[/size][/b][br][br]Let [math]f(x,y,z), g(x,y,z)[/math] and [math]h(x,y,z)[/math] be functions defined on a domain [math]R[/math] in [math]\mathbb{R}^3[/math]. A [b]vector field[/b] on [math]R[/math] is a vector-valued function [math]\vec{F}:R\to \mathbb{R}^3[/math] such that for any [math](x,y,z)[/math] in [math]R[/math],[br][br][math]\vec{F}(x,y,z)=\langle f(x,y,z), g(x,y,z), h(x,y,z) \rangle = f(x,y,z)\vec{i}+ g(x,y,z)\vec{j}+h(x,y,z)\vec{k}[/math][br][br]A vector field [math]\vec{F}=\langle f,g,h\rangle[/math] is continuous (or differentiable) on [math]R[/math] if [math]f, g[/math] and [math]h[/math] are continuous (or differentiable) on [math]R[/math].[br][br]The applet below illustrates a vector field [math]\vec{F}[/math] in 3D space - a vector (arrow) is assigned to every point in the cuboid [math][a,b]\times [a,b]\times [a,b][/math]. [br][br]You can try the following examples of vector fields:[br][br][list=1][*][math]\vec{F}(x,y,z)=\langle x, y, e^{-z}\rangle[/math] for [math]z\geq 0[/math][/*][*][math]\vec{F}(x,y,z)=\langle 0,0,16-x^2-y^2\rangle[/math] for [math]x^2+y^2\leq 16[/math] (a flow through the cylinder)[/*][*][math]\vec{F}(x,y,z)=\langle -y,x,0\rangle[/math] (a rotation field)[/*][*][math]\vec{F}(x,y,z)=\frac 1{|\langle x,y,z \rangle|}\langle x,y,z \rangle[/math], where [math]\langle x,y,z \rangle\ne\langle 0,0,0 \rangle[/math] (a radial vector field)[/*][/list]

[b][size=150]Gradient fields[/size][/b][br][br]Let [math]\phi(x,y)[/math] be a differentiable function on a domain in [math]\mathbb{R}^2[/math] . Then[br][br][math]\vec{F}(x,y)=\nabla \phi (x,y)=\langle \phi_x(x,y), \psi_y(x,y) \rangle[/math][br][br]is called a [b]gradient field[/b] in 2D space, and [math]\phi[/math] is the [b]potential functions[/b] for [math]\vec{F}[/math].[br][br]Similarly, if [math]\phi(x,y,z)[/math] is a differentiable function on a domain in [math]\mathbb{R}^3[/math], then[br][br][math]\vec{F}(x,y,z)=\nabla \phi(x,y,z)=\langle \phi_x(x,y,z), \phi_y(x,y,z), \phi_z(x,y,z) \rangle[/math][br][br]is a [b]gradient field[/b] in 3D space and [math]\phi[/math] is its potential function.[br][br][br][u]Remark[/u]: If [math]\phi[/math] is a potential function for a gradient field, then [math]\phi+C[/math] is also a potential function for that gradient field for any constant [math]C[/math].[br][br][br]Consider the graph of the potential function [math]z=\phi(x,y)[/math]. We already know that the gradient field [math]\nabla \psi[/math] at [math](x,y)[/math] is always pointing towards the direction of the steepest slope on the graph. Moreover, the gradient field is orthogonal to the level curve of the potential function.[br][br][u]Definition[/u]: A vector field [math]\vec{F}[/math] (in two or three dimension) is called a [b]conservative field[/b] if there exists a function [math]\phi[/math] such that [math]\vec{F}=\nabla \phi[/math][br][br][br][u]Example[/u]: Let [math]\vec{F}(x,y,z)=\frac{C}{(x^2+y^2+z^2)^{\frac 32}}\langle x, y, z \rangle[/math] be a radial vector field in 3D space, where [math]C[/math] is a constant. Show that it is a conservative field.[br][br]([u]Note[/u]: This vector field is called an [b]inverse-square field[/b]. In physics, electric field, gravitational field are all inverse-square fields.)[br][br][u]Answer[/u]:[br][br]Define [math]\phi(x,y,z)=-\frac{C}{\sqrt{x^2+y^2+z^2}}[/math].[br][br]Then we have[br][br][math]\nabla \phi=\langle \phi_x, \phi_y, \phi_z \rangle[/math][br][math]=\left\langle \frac{Cx}{(x^2+y^2+z^2)^{\frac 32}}, \frac{Cy}{(x^2+y^2+z^2)^{\frac 32}}, \frac{Cz}{(x^2+y^2+z^2)^{\frac 32}}\right\rangle[/math][br][math]=\frac{C}{(x^2+y^2+z^2)^{\frac 32}}\langle x, y, z \rangle=\vec{F}[/math] [br][br]In later chapter, we will learn how to determine whether a vector field is conservative or not.[br]