9  Vector Calculus

9.1 Vector Functions

• A parametric curve in space is a set of ordered triples \((x,y,z),\) \(~\)where

  \[x=f(t), \;y=g(t), \;z=h(t)\]

  are continuous on an interval: \(~a \leq t \leq b\)

• The following

  \[\mathbf{r}(t) =\left\langle f(t), \,g(t), \,h(t) \right\rangle = f(t)\,\mathbf{i} +g(t)\,\mathbf{j} +h(t)\,\mathbf{k}\]

  are vector-valued functions

• Limits

  • \(\displaystyle\lim_{t \to a} \mathbf{r}(t)=\left\langle \lim_{t \to a} f(t), \;\lim_{t \to a} g(t), \;\lim_{t \to a} h(t) \right\rangle\)

  • If \(~\)\(\displaystyle\lim_{t\to a} \mathbf{r}_1(t)=\mathbf{L}_1\) and \(~\)\(\displaystyle\lim_{t\to a} \mathbf{r}_2(t)=\mathbf{L}_2\), \(~\)then

    \(\displaystyle\lim_{t\to a} c\mathbf{r}_1(t)=c\mathbf{L}_1\)

    \(\displaystyle\lim_{t\to a} \left[\mathbf{r}_1(t) +\mathbf{r}_2(t) \right] =\mathbf{L}_1 +\mathbf{L}_2\)

    \(\displaystyle\lim_{t\to a} \left[\mathbf{r}_1(t) \cdot \mathbf{r}_2(t) \right] =\mathbf{L}_1 \cdot \mathbf{L}_2\)

• Continuity

  • A vector function \(\mathbf{r}\) is said to be continuous at \(t=a\) \(\,\)if

    \(\displaystyle\,\mathbf{r}(a) \;\text{ is defined, }\phantom{\frac{1}{1}}\)

    \(\displaystyle\lim_{t\to a} \mathbf{r}(t)\) exists, and

    \(\displaystyle\lim_{t\to a} \mathbf{r}(t) = \mathbf{r}(a)\)

• Derivatives

  • The derivative of a vector function \(\mathbf{r}\) is

    \(\displaystyle\phantom{xx}\mathbf{r}'(t) = \lim_{\Delta t\to 0} \frac{1}{\Delta t} \left[ \mathbf{r}(t +\Delta t) -\mathbf{r}(t) \right]\)

    for all \(\,t\,\) for which the limit exists

  • If \(~\mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle,\) \(\,\)where \(~f\), \(g\), and \(h\,\) are differentiable, then

    \(\phantom{xx}\mathbf{r}'(t) = \left\langle f'(t), g'(t), h'(t) \right\rangle\)

  • When \(\mathbf{r}\) have continuous first derivative and \(\mathbf{r}'(t)\neq\mathbf{0}~\) for all \(~t~\) in the open interval \((a,b),\) \(\,\)then \(\mathbf{r}\) is said to be a smooth function

    

  • In the case of the second derivative, \(\,\)we have

    \(\phantom{xx} \mathbf{r}''(t) = \left\langle f''(t), g''(t), h''(t) \right\rangle = f''(t)\mathbf{i} +g''(t)\mathbf{j} +h''(t)\mathbf{k}\)

  • Chain rule

    If \(\mathbf{r}\) is a differentiable vector function and \(s=u(t)\) is a differentiable scalar function, then the derivative of \(\mathbf{r}(s)\) with respect to \(t\) is

    \(\displaystyle\phantom{xx}\frac{d\mathbf{r}}{dt} =\frac{d\mathbf{r}}{ds} \frac{ds}{dt} =\mathbf{r}'(s) u'(t)\)

  • Rules of Differentiation

    Let \(\mathbf{r}_1(t)\) and \(\mathbf{r}_2(t)\) be differentiable vector functions and \(u(t)\) a differentiable scalar function

    • \(\displaystyle \frac{d}{dt} \left[ \mathbf{r}_1(t) +\mathbf{r}_2(t) \right] = \mathbf{r}_1'(t) +\mathbf{r}_2'(t)\)

    • \(\displaystyle \frac{d}{dt} \left[ u(t)\mathbf{r}_1(t) \right] = u(t)\mathbf{r}_1'(t) +u'(t)\mathbf{r}_1(t)\)

    • \(\displaystyle \frac{d}{dt} \left[ \mathbf{r}_1(t) \cdot \mathbf{r}_2(t) \right] = \mathbf{r}_1(t)\cdot\mathbf{r}_2'(t) +\mathbf{r}_1'(t)\cdot\mathbf{r}_2(t)\)

    • \(\displaystyle \frac{d}{dt} \left[ \mathbf{r}_1(t) \times \mathbf{r}_2(t) \right] = \mathbf{r}_1(t)\times\mathbf{r}_2'(t) +\mathbf{r}_1'(t)\times\mathbf{r}_2(t)\)

• Integrals

  \(\phantom{xx}\displaystyle\int \mathbf{r}(t) \,dt = \left[ \int f(t) \,dt \right]\mathbf{i} +\left[ \int g(t) \,dt \right]\mathbf{j} +\left[ \int h(t) \,dt \right]\mathbf{k}\)

• Length of a Space Curve

  If \(~\mathbf{r}(t) = f(t)\mathbf{i} +g(t)\mathbf{j} +h(t)\mathbf{k}\,\) is a smooth function, \(\,\)then it can be shown that the length of the smooth curve traced by \(\,\mathbf{r}\,\) is given by

  \(\phantom{xx}\displaystyle s = \int_a^b \sqrt{\left[ f'(t) \right]^2 +\left[ g'(t) \right]^2 +\left[ h'(t) \right]^2}\,dt =\int_a^b \left\| \mathbf{r}'(t) \right\|\,dt\)

9.2 Motion on a Curve

• Position

  Suppose a body moves along a curve \(C\) so that its position at time \(t\) is given by the vector function

  \(\phantom{xx}\mathbf{r}(t) = f(t)\mathbf{i} +g(t)\mathbf{j} +h(t)\mathbf{k}\)

• Velocity and Acceleration

  If \(\,f\), \(g\), and \(h\) have second derivatives, \(\,\) then the vectors

  \(\phantom{xx}\begin{aligned} \mathbf{v}(t) &= \mathbf{r}'(t) = f'(t)\mathbf{i} +g'(t)\mathbf{j} +h'(t)\mathbf{k} \\ \mathbf{a}(t) &= \mathbf{r}''(t) = f''(t)\mathbf{i} +g''(t)\mathbf{j} +h''(t)\mathbf{k} \end{aligned}\)

  are called the velocity and acceleration of the particle, respectively

  • The scalar function \(\left\| \mathbf{v}(t)\right\|\) is the speed of the particle

  • The speed is related to arc length \(s\) by \(s'(t) = \left\| \mathbf{v}(t) \right\|\)

  • If a particle moves with a constant speed \(c\), \(\,\) then its acceleration vector is perpendicular to the velocity vector \(\mathbf{v}\)

  • To see this, note that \(\mathbf{v}\cdot\mathbf{v}=c^2\). \(~\) We differentiate both sides with respect to \(t\) and obtain

    \[{\frac{d}{dt}(\mathbf{v} \cdot \mathbf{v}) = \mathbf{v}\cdot\frac{d\mathbf{v}}{dt} +\frac{d\mathbf{v}}{dt} \cdot \mathbf{v} =2\mathbf{v}\cdot\frac{d\mathbf{v}}{dt}=0}\]

    Thus, \(\displaystyle\,\frac{d\mathbf{v}}{dt} \cdot \mathbf{v}=0\) \(\text{ }\)or\(\text{ }\) \(\mathbf{a}(t) \cdot \mathbf{v}(t)=0~\,\) for all \(\,t\)

• Centripetal Acceleration

  For circular motion in the plane, described by \(\mathbf{r}(t) = r_0 \cos\omega t \,\mathbf{i} +r_0\sin\omega t \,\mathbf{j},\) \(\,\)it is evident that

  \[\mathbf{r}''=-\omega^2\mathbf{r}\]

  • This means that the acceleration vector \(\mathbf{a}(t)=\mathbf{r}''(t)\) points in the direction opposite to that of the position vector \(\mathbf{r}(t)\). \(\,\) We then say \(~\mathbf{a}(t)\) is centripetal acceleration

9.3 Curvature and Components of Acceleration

• We know that \(\mathbf{r}'(t)\) is a tangent vector to the curve \(C\), and consequently

  \[\mathbf{T}(t) = \frac{\mathbf{r}'(t)}{\left\| \mathbf{r}'(t) \right\|}\]

  is a unit tangent

\[ \begin{aligned} \frac{d\mathbf{r}}{dt} &=\frac{d\mathbf{r}}{ds} \frac{ds}{dt}\;\text{ and so }\\[5pt] \frac{d\mathbf{r}}{ds} = \frac{\displaystyle\frac{d\mathbf{r}}{dt}} {\displaystyle\frac{ds}{dt}} &=\frac{\mathbf{r}'(t)}{\left\| \mathbf{r}'(t) \right\|} = \mathbf{T}(t) \end{aligned}\]

• Let \(\,\mathbf{r}(t)\,\) be a vector function defining a smooth curve \(C\). \(\,\)If \(s\) is the arc length parameter and \(\mathbf{T}\) is the unit tangent vector, \(\,\)then the curvature of \(C\) at a point is

  \[\kappa=\left\|\frac{d\mathbf{T}}{ds}\right\|\]

• Using the chain rule, \(\,\)we can write

  \[\frac{d\mathbf{T}}{dt}=\frac{d\mathbf{T}}{ds}\frac{ds}{dt} \;\;\Rightarrow\;\; \frac{d\mathbf{T}}{ds}=\frac{\displaystyle\frac{d\mathbf{T}}{\displaystyle dt}}{\frac{\displaystyle ds}{\displaystyle dt}}\]

  In other words, \(\,\)curvature is given by \(\,\kappa(t)=\frac{\displaystyle\left\|\mathbf{T}'(t)\right\|}{\displaystyle\left\|\mathbf{r}'(t)\right\|}\)

• Tangential and Normal Components of Acceleration

  • The velocity of the particle on \(C\) is \(~\mathbf{v}(t)=\mathbf{r}'(t)\), \(\,\)whereas its speed is \(\displaystyle\frac{ds}{dt}=v=\| \mathbf{v}(t) \|\). \(\,\) Thus, \[\mathbf{v}(t)=v\mathbf{T}\]

  • Differentiating this last expression with respect to \(t\) gives acceleration

    \[\mathbf{a}(t)=v\frac{d\mathbf{T}}{dt}+\frac{dv}{dt}\mathbf{T}\]

  • It follows from the differentiation of \(\mathbf{T}\cdot\mathbf{T}=1\,\) that \(\displaystyle\,\mathbf{T}\cdot \frac{d\mathbf{T}}{dt}=0\). \(\,\)If \(\displaystyle\,\left\|\frac{d\mathbf{T}}{dt}\right\| \neq 0\), \(\,\)the vector

    \[\mathbf{N}(t) = \frac{\frac{\displaystyle d\mathbf{T}}{\displaystyle dt}}{\left\| \frac{\displaystyle d\mathbf{T}}{\displaystyle dt}\right\|}\]

    is a unit normal to the curve \(C\). \(\,\)The vector \(\mathbf{N}\) is also called the principal normal

  • Since curvature is \(\displaystyle\,\kappa = \frac{\left\|\frac{d\mathbf{T}}{dt} \right\|}{v}\), \(\,\) it follows that \(\displaystyle\frac{d\mathbf{T}}{dt}=\kappa v \mathbf{N}\). \(\,\)Thus

    \[\mathbf{a}(t)=\kappa v^2 \mathbf{N} +\frac{dv}{dt}\mathbf{T} =a_N \mathbf{N} +a_T\mathbf{T}\]

• Binormal: \(\,\mathbf{B}(t)=\mathbf{T}(t) \times \mathbf{N}(t)\)

• Formulas for \(a_T\), \(a_N\), and Curvature

  \[ \begin{aligned} \mathbf{v}\cdot\mathbf{a} &= a_N(v\mathbf{T}\cdot\mathbf{N}) +a_T(v\mathbf{T}\cdot\mathbf{T})=a_T v\\[5pt] &\Rightarrow \; a_T =\frac{dv}{dt} =\frac{\mathbf{v}\cdot\mathbf{a}}{\| \mathbf{v} \|} =\frac{\mathbf{r}'(t) \cdot \mathbf{r}''(t)}{\| \mathbf{r}'(t) \|} \\ \\ \mathbf{v}\times\mathbf{a} &= a_N(v\mathbf{T}\times\mathbf{N}) +a_T(v\mathbf{T}\times\mathbf{T}) =a_N v\mathbf{B}\\[5pt] &\Rightarrow \; a_N =\kappa v^2 =\frac{\|\mathbf{v}\times\mathbf{a}\|}{\| \mathbf{v} \|} =\frac{\| \mathbf{r}'(t) \times \mathbf{r}''(t) \|}{\| \mathbf{r}'(t) \|} \\ \\ \kappa(t)&=\frac{\| \mathbf{r}'(t)\times\mathbf{r}''(t) \|}{\| \mathbf{r}'(t) \|^3} \end{aligned}\]

• Radius of Curvature

  \[\rho=\frac{1}{\kappa}\]

\(~\)

9.4 Partial Derivatives

• Functions of Two Variables

• Level Curves

• Functions of Three Variables, Level Surfaces

  \[w=F(x,y,z) \;\Rightarrow \; c=F(x,y,z)\]

  ex) \(~\displaystyle w=\frac{x^2+y^2}{z}\)

• Partial Derivatives

  \(\phantom{xx} \begin{aligned} \frac{\partial z}{\partial x} &=\lim_{\Delta x \to 0} \frac{f(x+\Delta x, y) -f(x,y)}{\Delta x} = f_x \\ \frac{\partial z}{\partial y} &=\lim_{\Delta y \to 0} \frac{f(x, y+\Delta y) -f(x,y)}{\Delta y} = f_y \\ \frac{\partial^2 z}{\partial x^2} &=\frac{\partial}{\partial x}\left( \frac{\partial z}{\partial x} \right) = f_{xx} \\ \frac{\partial^2 z}{\partial x \partial y} &=\frac{\partial}{\partial x} \left( \frac{\partial z}{\partial y} \right) = f_{xy} \end{aligned}\)

• Chain Rule

  If \(z=f(u,v)\) is differentiable and \(u=g(x,y)\) and \(v=h(x,y)\) have continuous first partial derivatives, \(\,\)then

  \[ \begin{aligned} \frac{\partial z}{\partial x}&= \frac{\partial z}{\partial u}\frac{\partial u}{\partial x} +\frac{\partial z}{\partial v}\frac{\partial v}{\partial x}\\ \frac{\partial z}{\partial y}&= \frac{\partial z}{\partial u}\frac{\partial u}{\partial y} +\frac{\partial z}{\partial v}\frac{\partial v}{\partial y} \end{aligned}\]

  If \(z=f(u,v)\) is differentiable and \(u=g(t)\,\) and \(v=h(t)\) are differentiable functions of a single variable \(t\), \(\,\)the ordinary derivative \(\displaystyle\frac{dz}{dt}\) is

  \[\frac{dz}{dt}=\frac{\partial z}{\partial u}\frac{du}{dt}+\frac{\partial z}{\partial v}\frac{dv}{dt}\]

\(~\)

9.5 Directional Derivatives

• Vector Differential Operator

  \(\phantom{xx} \begin{aligned} \nabla &= \mathbf{i} \frac{\partial }{\partial x} +\mathbf{j}\frac{\partial }{\partial y} \\ \nabla &= \mathbf{i}\frac{\partial }{\partial x} +\mathbf{j}\frac{\partial }{\partial y} +\mathbf{k}\frac{\partial }{\partial z} \end{aligned}\)

• Gradient of a Function

  \(\phantom{xx} \begin{aligned} \nabla f &= \mathbf{i} \frac{\partial f}{\partial x} +\mathbf{j}\frac{\partial f}{\partial y} \\ \nabla F &= \mathbf{i}\frac{\partial F}{\partial x} +\mathbf{j}\frac{\partial F}{\partial y} +\mathbf{k}\frac{\partial F}{\partial z} \end{aligned}\)

\(~\)

• Generalization of Partial Differentiation

  The directional derivative of \(z=f(x,y)\,\) in the direction of a unit vector \(\mathbf{u}\) \(\,\)is

  \[ \begin{aligned} D_\mathbf{u} f(x,y) &= \lim_{h\to 0} \frac{f(x+\Delta x, \,y+\Delta y) -f(x,y)}{h} \\ &= \lim_{h\to 0} \frac{f(x+h\cos\theta, \,y+h\sin\theta) -f(x,y)}{h} = \nabla f(x,y)\cdot \mathbf{u} \end{aligned}\]

• Partial Proof \(\,\) Let \(x\), \(y\), and \(\theta\) \(\,\)be fixed so that

  \[g(t)=f(x+t\cos\theta,\,y+t\sin\theta)\]

  is a function of one variable. \(\,\)We wish to compare the value of \(g'(0)\), \(\,\)which is found by two different methods

  • First, \(\,\)by the definition of a derivative,

  \[{g'(0)=\lim_{h\to 0} \frac{g(0+h)-g(0)}{h} =\lim_{h\to 0}\frac{f(x+h\cos\theta, \,y+h\sin\theta)-f(x,y)}{h} }\]

  • Second, \(\,\)by the chain rule,

    \[ \begin{aligned} g'(t) &= f_1(x+t\cos\theta,\,y+t\sin\theta)\frac{d}{dt}(x+t\cos\theta) \\ &\qquad +f_2(x+t\cos\theta,\,y+t\sin\theta)\frac{d}{dt}(x+t\sin\theta) \\ &= f_1(x+t\cos\theta,\,y+t\sin\theta)\cos\theta +f_2(x+t\cos\theta,\,y+t\sin\theta)\sin\theta \\ \text{ } \\ &\big\Downarrow\;\; t\to 0 \\ \text{ } \\ g'(0) &= f_x(x,y)\cos\theta+f_y(x,y)\sin\theta \\[5pt] &=\left[\,f_x(x,y)\,\mathbf{i}+f_y(x,y)\,\mathbf{j} \,\right]\cdot \left(\cos\theta\,\mathbf{i}+\sin\theta\,\mathbf{j}\right) =\nabla f(x,y)\cdot \mathbf{u} \end{aligned}\]

• For a function \(w=F(x,y,z)\), \(~\)the directional derivative is defined by

  \[{\scriptsize \displaystyle D_\mathbf{u} F(x,y,z) = \lim_{h\to 0} \frac{F(x+h\cos\alpha, \,y+h\cos\beta, \,z +h\cos\gamma) -F(x,y,z)}{h} =\nabla F(x,y,z)\cdot \mathbf{u}}\]

• Maximum Value of the Directional Derivative

  \[ \begin{aligned} D_\mathbf{u}\,f &= \| \nabla f\| \| \mathbf{u} \| \cos\phi =\| \nabla f\| \cos\phi \\[5pt] &\,\Rightarrow\, \color{red}{-\| \nabla f\| \leq D_\mathbf{u} \,f \leq \| \nabla f \|} \end{aligned}\]

• The gradient vector \(\nabla f\) points in the direction in which \(f\) increases most rapidly, \(\,\)whereas \(-\nabla f\) points in the direction of the most decrease of \(\,f\)

\(~\)

9.6 Tangent Planes and Normal Lines

• \(\nabla f\) is orthogonal to the level curve at \(P\)

• The derivative of \(\,f\left(x(t), \,y(t)\right)=c\,\) with respect to \(\,t\,\) is

  \[{\frac{\partial f}{\partial x}\frac{dx}{dt} +\frac{\partial f}{\partial y}\frac{dy}{dt}=0 =\left( \frac{\partial f}{\partial x}\mathbf{i} +\frac{\partial f}{\partial y} \mathbf{j} \right) \cdot \left( \frac{dx}{dt}\mathbf{i} +\frac{dy}{dt} \mathbf{j} \right) =\nabla f \cdot \mathbf{r}'}\]

• The derivative of \(\,F\left(x(t),y(t),z(t)\right)=c\,\) implies that

  \[ \begin{aligned} \frac{\partial F}{\partial x}&\frac{dx}{dt} +\frac{\partial F}{\partial y}\frac{dy}{dt} +\frac{\partial F}{\partial z}\frac{dz}{dt}=0 \\ &=\left( \frac{\partial F}{\partial x}\mathbf{i} +\frac{\partial F}{\partial y} \mathbf{j} +\frac{\partial F}{\partial z} \mathbf{k} \right) \cdot \left( \frac{dx}{dt}\mathbf{i} +\frac{dy}{dt} \mathbf{j} +\frac{dz}{dt} \mathbf{k} \right) =\nabla F \cdot \mathbf{r}' \end{aligned}\]

• \(\nabla F\) is normal to the level surface at \(P\)

• Let \(\,P(x_0,y_0,z_0)\) be a point on the graph of \(F(x,y,z)=c\). \(\,\)The tangent plane at \(P\) is

  \[\\ F_x(x_0,y_0,z_0)(x-x_0) +F_y(x_0,y_0,z_0)(y-y_0) +F_z(x_0,y_0,z_0)(z-z_0)=0 \\\]

  The line containing \(P(x_0,y_0,z_0)\) \(\,\)that is parallel to \(\nabla F(x_0,y_0,z_0)\) \(\,\)is called the normal line to the surface at \(P\)

\(~\)

9.7 Gradient, Curl and Divergence

• Vector Fields - Vector functions of two or three variables

  \[ \begin{aligned} \mathbf{f}(x,y) &= P(x,y) \mathbf{i} +Q(x,y)\mathbf{j}\\ \mathbf{f}(x,y,z) &= P(x,y,z) \mathbf{i} +Q(x,y,z)\mathbf{j} +R(x,y,z)\mathbf{k} \end{aligned}\]

• The del operator combined with a scalar function \(\,\phi(x,y,z)\) \(\,\)produces a vector field

  \[\mathbf{f}(x,y,z)=\nabla \phi = \mathrm{grad}\, f =\frac{\partial \phi}{\partial x}\mathbf{i} +\frac{\partial \phi}{\partial y}\mathbf{j} +\frac{\partial \phi}{\partial z}\mathbf{k} \;\Rightarrow\; \delta_{ij} \mathbf{e}_i \frac{\partial \phi}{\partial x_j}\]

  called the gradient of \(\phi\), \(\,\)where \(\,\delta_{ij}\) \(\,\)is Kronecker delta

• The curl of a vector field \(~\mathbf{f}=P\,\mathbf{i} +Q\,\mathbf{j} +R\,\mathbf{k}\,\) is the vector field

  \[\nabla \times \mathbf{f}= \mathrm{curl}\, \mathbf{f}= \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k}\\ \frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z} \\ P & Q & R \end{vmatrix} \; \Rightarrow \; \varepsilon_{ijk} \mathbf{e}_i \frac{\partial}{\partial x_j} f_k\]

  where \(\,\varepsilon_{ijk}\) is Levi-Civita Symbol

• The divergence of a vector field \(~\mathbf{f}=P\,\mathbf{i} +Q\,\mathbf{j} + R\,\mathbf{k}~\) is the scalar function

  \[\nabla \cdot \mathbf{f}= \mathrm{div} \,\mathbf{f}= \frac{\partial P}{\partial x} +\frac{\partial Q}{\partial y} +\frac{\partial R}{\partial z} \;\Rightarrow\;\delta_{ij} \frac{\partial}{\partial x_i} f_j\]

• Two important properties and plus alpha

  \(\phantom{xx} \begin{aligned} \nabla\times\nabla f &= \mathrm{curl}(\mathrm{grad}\, f) = \mathbf{0}\\ &\Rightarrow \; \varepsilon_{ijk} \mathbf{e}_i \frac{\partial}{\partial x_j} \frac{\partial f}{\partial x_k} = 0 \,\mathbf{e}_i=\mathbf{0} \end{aligned}\)

  \(\phantom{xx} \begin{aligned} \nabla \cdot(\nabla\times\mathbf{f}) &= \mathrm{div}(\mathrm{curl} \,\mathbf{f}) =0 \\ &\Rightarrow \; \delta_{ij} \frac{\partial}{\partial x_i} \varepsilon_{jlm} \frac{\partial}{\partial x_l} f_m = \varepsilon_{ilm} \frac{\partial^2}{\partial x_i \partial x_l} f_m=0 \end{aligned}\)

  \(\displaystyle \phantom{xx} \nabla \cdot \nabla f =\nabla^2 f \; \Rightarrow \; \delta_{ij} \frac{\partial}{\partial x_i} \frac{\partial f}{\partial x_j} =\frac{\partial^2 f}{\partial x_i^2}\)

  \(\displaystyle \phantom{xx} \nabla \cdot \nabla \mathbf{f} =\nabla^2 \mathbf{f} \; \Rightarrow \; \delta_{ij} \frac{\partial}{\partial x_i} \mathbf{e}_m \frac{\partial f_m}{\partial x_j} =\mathbf{e}_m \frac{\partial^2 f_m}{\partial x_i^2}\)

  \(\displaystyle \phantom{xx} \begin{aligned} \nabla \times (\nabla \times \mathbf{f}) &= \nabla (\nabla \cdot \mathbf{f}) -\nabla^2 \mathbf{f} \\ &\Rightarrow \; \varepsilon_{ijk} \mathbf{e}_i \frac{\partial}{\partial x_j} \varepsilon_{klm} \frac{\partial}{\partial x_l}f_m =(\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}) \mathbf{e}_i \frac{\partial}{\partial x_j} \frac{\partial}{\partial x_l}f_m \\ &\phantom{xx} =\mathbf{e}_i \frac{\partial}{\partial x_i} \frac{\partial f_j}{\partial x_j} -\mathbf{e}_i \frac{\partial^2 f_i}{\partial x_j^2} \end{aligned}\)

• Product Rules with \(\nabla\)

  \(\phantom{xx} \begin{aligned} \nabla (fg) &= f \nabla g + g \nabla f \\ \nabla (\mathbf{u} \cdot \mathbf{v}) &= \mathbf{u} \times (\nabla \times \mathbf{v}) +\mathbf{v} \times (\nabla \times \mathbf{u}) +(\mathbf{u} \cdot \nabla) \mathbf{v} +(\mathbf{v} \cdot \nabla) \mathbf{u} \\ &= \mathbf{v} \cdot \nabla \mathbf{u} +\mathbf{u} \cdot \nabla \mathbf{v} \\ \nabla \cdot (f \mathbf{u}) &= f (\nabla \cdot \mathbf{u}) +\mathbf{u} \cdot \nabla f \\ \nabla \cdot (\mathbf{u} \times \mathbf{v}) &= \mathbf{v} \cdot (\nabla \times \mathbf{u}) -\mathbf{u} \cdot (\nabla \times \mathbf{v}) \\ \nabla \times (f \mathbf{u}) &= \nabla f \times \mathbf{u} +f (\nabla \times \mathbf{u}) \\ \nabla \times (\mathbf{u} \times \mathbf{v}) &= \mathbf{u} (\nabla \cdot \mathbf{v}) -\mathbf{v} (\nabla \cdot \mathbf{u}) +(\mathbf{v} \cdot \nabla) \mathbf{u} -(\mathbf{u} \cdot \nabla) \mathbf{v} \end{aligned}\)

• Generalized Curvlinear Coorindates: \(\,\) \(\mathbf{q}_i\), \(\,q_i\), \(\,h_i\)

  \[ \begin{aligned} \nabla f \; &= \mathbf{q}_1 \frac{1}{h_1} \frac{\partial f}{\partial q_1} + \mathbf{q}_2 \frac{1}{h_2} \frac{\partial f}{\partial q_2} + \mathbf{q}_3 \frac{1}{h_3} \frac{\partial f}{\partial q_3} \\ \\ \nabla \cdot \mathbf{u} \; &= \frac{1}{h_1 h_2 h_3} \left[ \frac{\partial }{\partial q_1} \left( u_1h_2h_3 \right) + \frac{\partial }{\partial q_2} \left( u_2h_1h_3 \right) + \frac{\partial }{\partial q_3} \left( u_3h_1h_2 \right)\right] \\ \\ \nabla^2 f\; & = \frac{1}{h_1 h_2 h_3} \left[ \frac{\partial }{\partial q_1} \left( \frac{h_2 h_3}{h_1} \frac{\partial f}{\partial q_1} \right) + \frac{\partial }{\partial q_2} \left( \frac{h_1 h_3}{h_2} \frac{\partial f}{\partial q_2} \right) + \frac{\partial }{\partial q_3} \left( \frac{h_1 h_2}{h_3} \frac{\partial f}{\partial q_3} \right)\right]\\ \\ \nabla \times \mathbf{u} \; &=\frac{1}{h_1 h_2 h_3} \begin{vmatrix} h_1 \mathbf{q}_1 & h_2 \mathbf{q}_2 & h_3 \mathbf{q}_3\\ \displaystyle\frac{\partial }{\partial q_1} & \displaystyle\frac{\partial }{\partial q_2} & \displaystyle\frac{\partial }{\partial q_3} \\ h_1 u_1 & h_2 u_2 & h_3 u_3 \end{vmatrix} \end{aligned}\]

  • Cylindrical Coordinates

    \(\phantom{xx} q_1=r, \;q_2=\theta, \; q_3=z\)

    \(\phantom{xx} h_1=h_r=1, \;\;h_2=h_\theta=r, \;\;h_3=h_z=1\)

  • Spherical Coordinates

    \(\phantom{xx} q_1=\rho, \;q_2=\phi, \; q_3=\theta\)

    \(\phantom{xx} h_1=h_\rho=1, \;\;h_2=h_\phi=\rho, \;\;h_3=h_\phi=\rho \sin\phi\)

• Physical Interpretations

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9.8 Line Integrals

• Integration of a function defined along a curve

• Definite Integral

  \[\lim_{\|P\|\to 0} \sum_{k=1}^n f(x_k^*)\Delta x_k = \int_a^b f(x)\,dx\]

• Line Integrals in the Plane

\[ \begin{aligned} \lim_{\| P \|\to 0} \sum_{k=1}^n G(x_k^*, y_k^*)\,\Delta x_k &= \int_C G(x,y)\,dx\\ \lim_{\| P \|\to 0} \sum_{k=1}^n G(x_k^*, y_k^*)\,\Delta y_k &=\int_C G(x,y)\,dy\\ \lim_{\| P \|\to 0} \sum_{k=1}^n G(x_k^*, y_k^*)\,\Delta s_k &= \int_C G(x,y)\,ds \end{aligned}\]

• Method of Evaluation

  • Curve Defined Parametrically, \(\;x=f(t)\), \(\;y=g(t)\), \(\;a\leq t \leq b\)

    \[ \begin{aligned} \int_C G(x,y)\,dx &=\int_a^b G(\,f(t), g(t)) \,f'(t)\,dt \\ \int_C G(x,y)\,dy &=\int_a^b G(\,f(t), g(t)) \,g'(t)\,dt \\ \int_C G(x,y)\,ds &=\int_a^b G(\,f(t), g(t)) \,\sqrt{[f'(t)]^2 +[g'(t)]^2}\,dt \end{aligned}\]

  • Curve Defined by an Explicit Function, \(\;y=f(x)\), \(\;a\leq x \leq b\)

    \[ \begin{aligned} \int_C G(x,y)\,dx &=\int_a^b G(x, f(x)) \,dx \\ \int_C G(x,y)\,dy &=\int_a^b G(x, f(x)) \,f'(x)\,dx \\ \int_C G(x,y)\,ds &=\int_a^b G(x, f(x)) \,\sqrt{1 +[f'(x)]^2}\,dx \end{aligned}\]

• Notation

  \[{\displaystyle\int_C P\,dx +\int_C Q\,dy \,\Rightarrow \, \int_C P\,dx +Q \,dy}\]

  \[{\displaystyle\int_{-C} P\,dx +Q\,dy = -\int_C P\,dx +Q\,dy}\]

• A line integral along a closed curve \(C\)

  \[\oint_C P \,dx +Q \,dy\]

• Line Integrals in Space

  \[{\int_C G(x,y,z)\,ds = \int_a^b G(\,f(t), g(t),h(t)) \,\sqrt{[f'(t)]^2 +[g'(t)]^2 +[h'(t)]^2}\,dt}\]

• Suppose the vector-valued function

  \[\mathbf{f}(x,y,z)=P(x,y,z)\mathbf{i} +Q(x,y,z)\mathbf{j} +R(x,y,z)\mathbf{k}\]

  is defined along a curve \(C\) and

  \[d\mathbf{r}=dx\mathbf{i} +dy\mathbf{j} +dz\mathbf{k}\,\]

  is the displacement vector of points on \(C\), \(\,\)then

  \[\int_C \mathbf{f}\cdot \,d\mathbf{r} =\int_C P(x,y,z) \, dx +Q(x,y,z) \, dy +R(x,y,z) \, dz\]

• Work and Circulation \[W =\int_C \mathbf{f} \cdot d\mathbf{r} = \int_C \mathbf{f}\cdot \mathbf{T} \,ds \] \[\mathrm{Circulation}= \oint_C \mathbf{f} \cdot d\mathbf{r} = \oint_C \mathbf{f}\cdot \mathbf{T} \,ds \]

\(~\)

9.9 Independence of the Path

• The value of a line integral

  \[\int_C \mathbf{F}\cdot \,d\mathbf{r}\]

  depends on the path of integration

• Stated another way, \(\,\)if \(C_1\) and \(C_2\) are two different paths between the same points \(A\) and \(B\), \(\,\)then we expect that \[\int_{C_1} \mathbf{F}\cdot \,d\mathbf{r} \neq \int_{C_2} \mathbf{F}\cdot \,d\mathbf{r}\]

• Path Independence

  A line integral \(\displaystyle\int_C \mathbf{f}\cdot d\mathbf{r}~\) is independence of the path if

  \[\int_{C_1} \mathbf{f}\cdot d\mathbf{r}=\int_{C_2}\mathbf{f}\cdot d\mathbf{r}\]

  for any two paths \(C_1\) and \(C_2\) between \(A\) and \(B\)

• A vector function \(\mathbf{f}\) in 2- or 3-space is said to be conservative if \(\,\mathbf{f}\) can be written as the gradient of a scalar function \(\phi\). \(\,\)The function \(\phi\) is called a potential function for \(\,\mathbf{f}\)

• In other words, \(\,\mathbf{f}\) is conservative if there exists a function \(\phi\) such that \(\,\mathbf{f}=\nabla \phi\). \(\,\)A conservative vector field is also called a gradient vector field

• Fundamental Theorem

  If \(~\mathbf{f}(x,y,z)~\) is a conservative vector field in \(R\) and \(\phi\) is a potential function for \(\,\mathbf{f}\), \(\,\)then

  \[\int_C \mathbf{f}\cdot d\mathbf{r}=\int_C \nabla\phi \cdot d\mathbf{r}=\phi(B) -\phi(A)\]

  where \(A=(x(a), y(a), z(a))~\) and \(\,B=(x(b), y(b), z(b))\)

  

• Equivalent Concepts: \(\,\)In an open connected region \(R\), \(\displaystyle\int_C \mathbf{f}\cdot d\mathbf{r}\) is independent of the path \(C\)

  • if and only if the vector field \(\mathbf{f}\) is conservative in \(R\)

  • if and only if \(\displaystyle\oint_{C'} \mathbf{f}\cdot d\mathbf{r}=0\,\) for every closed path \(\,C'\) in \(R\)

• Test for a Conservative Field:

  • Suppose

    \[\mathbf{f}(x,y,z)=P(x,y,z)\mathbf{i} +Q(x,y,z)\mathbf{j} +R(x,y,z)\mathbf{k}\]

    is a conservative vector field in an open region of 3-space, and that \(P\), \(Q\), and \(R\) are continuous and have continuous first partial derivatives in that region

  • Then \(\,\mathbf{f}=\nabla \phi\,\) and \(\,\nabla\times\mathbf{f}=\nabla\times\nabla\phi=\mathbf{0}\), \(\,\)that is

    \[{\nabla\times\mathbf{f}=\left( \frac{\partial R}{\partial y} -\frac{\partial Q}{\partial z}\right)\mathbf{i} +\left( \frac{\partial P}{\partial z} -\frac{\partial R}{\partial x}\right)\mathbf{j} +\left( \frac{\partial Q}{\partial x} -\frac{\partial P}{\partial y}\right)\mathbf{k}=\mathbf{0}}\;\;\]

    \[~\;\Downarrow\]

    \[\;\;{\frac{\partial P}{\partial y} =\frac{\partial Q}{\partial x}, \;\frac{\partial P}{\partial z} =\frac{\partial R}{\partial x},\; \frac{\partial Q}{\partial z} =\frac{\partial R}{\partial y}}\]

    for all \(\,(x,y,z)\) in that region

\(~\)

9.10 Double Integrals

• The Double Integral

  Let \(\,f\) be a function of two variables defined on a closed region \(R\). Then the double integral of \(\,f\) over \(R\) \(\,\)is given by

  \[\lim_{\|P\|\to 0} \sum_{k=1}^n f\left(x_k^*, y_k^*\right) \Delta A_k = \iint_R f(x,y)\,dA\]

• Area and Volume

  \[ A=\iint_R dA \]

  \[ V=\iint_R f(x,y) \,dA \]

• Properties of Double Integrals

  \[ \iint_R kf(x,y) \,dA = k \iint_R f(x,y) \,dA \]

  \[ \iint_R f(x,y) \pm g(x,y) \,dA = \iint_R f(x,y) \,dA \pm \iint_R g(x,y) \,dA \]

  \[ \iint_R f(x,y) \,dA = \iint_{R_1} f(x,y) \,dA + \iint_{R_2} f(x,y) \,dA \]

  

• Evaluation of Double Integrals

  

  • If \(R\) is of Type I, \(\,\)then

    \[{\iint_R f(x,y) \,dA =\int_a^b\int_{g_1(x)}^{g_2(x)} f(x,y) \,dydx}\]

  • If \(R\) is of Type II, \(\,\)then

    \[{\iint_R f(x,y) \,dA =\int_c^d\int_{h_1(y)}^{h_2(y)} f(x,y) \,dxdy}\]

• Center of Mass and Moments of Inertia

  \[ \bar{x} = \frac{\displaystyle\iint_R x\rho(x,y)\,dA}{M} \, \;\textrm{ and } \; \bar{y}=\frac{\displaystyle\iint_R y\rho(x,y) \,dA}{M}\]

  \[ I_y=\iint_R x^2\rho(x,y) \, dA\;\textrm{ and } \; I_x=\iint_R y^2\rho(x,y) \,dA \]

\(~\)

9.11 Double Integrals in Polar Coordinates

• Polar Rectangles

  \[ \begin{aligned} \Delta A_k &=\frac{1}{2}(r_{k+1}^2 -r_k^2)\Delta \theta_k \\ &=\frac{1}{2}(r_{k+1} +r_k)(r_{k+1} -r_k) \Delta\theta_k \\[5pt] &=r_k^*\Delta r_k \Delta\theta_k \end{aligned}\]

• Double Integrals in Polar Coordinates

  \[ \begin{aligned} \lim_{\| P \|\to 0} \sum_{k=1}^n & f\left( r_k^*, \theta_k^* \right) r_k^* \Delta r_k \Delta\theta_k \\ & = \iint_R f(r,\theta) \,dA\\ & = \int_\alpha^\beta \int_{g_1(\theta)}^{g_2(\theta)} f(r,\theta)\,r \,dr \,d\theta \\ & =\int_a^b \int_{h_1(r)}^{h_2(r)} f(r,\theta)\,r \,d\theta \,dr \end{aligned}\]

\(~\)

9.12 Green’s Theorem

• Suppose that \(C\) is a piecewise-smooth simple closed curve bounding a simply connected region \(R\)

• If \(\,P\), \(Q\), \(\displaystyle\frac{\partial P}{\partial y}\), and \(\displaystyle\frac{\partial Q}{\partial x}\) are continuous on \(R\), \(\,\)then

  \[{\oint_C P \,dx +Q \,dy = \iint_R \left( \frac{\partial Q}{\partial x} -\frac{\partial P}{\partial y} \right)\, dA}\]

\(~\)

• Partial Proof

  \[ \begin{aligned} -\iint_R \frac{\partial P}{\partial y}\, dA &= -\int_a^b \int_{g_1(x)}^{g_2(x)} \frac{\partial P}{\partial y} \,dy dx =-\int_a^b [P(x,g_2(x)) -P(x,g_1(x))] \,dx\\ & =\int_a^b P(x,g_1(x))\,dx +\int_b^a P(x,g_2(x)) \,dx = \oint_C P(x,y) \,dx \end{aligned}\]

  \(~\)

  

• Region with Holes

  \[ \begin{aligned} \iint_R \left( \frac{\partial Q}{\partial x} -\frac{\partial P}{\partial y} \right)\, dA &=\iint_{R_1} \left( \frac{\partial Q}{\partial x} -\frac{\partial P}{\partial y} \right)\, dA +\iint_{R_2} \left( \frac{\partial Q}{\partial x} -\frac{\partial P}{\partial y} \right)\, dA\\ &=\oint_{C_1} P \,dx +Q \,dy +\oint_{C_2} P \,dx +Q \,dy =\oint_C P \,dx +Q \,dy \end{aligned}\]

  

\(~\)

Example \(\,\) Evaluate \(\displaystyle\oint_{C} \frac{-y}{x^2 +y^2}\, dx + \frac{x}{x^2 + y^2}\,dy\), \(\,\) where \(C=C_1 \cup C_2\) is the boundary of the shaded region \(R\)

\(~\)

Example \(\,\) Evaluate \(\displaystyle\oint_{C_1} \frac{-y}{x^2 +y^2}\, dx + \frac{x}{x^2 + y^2}\,dy\)

\(~\)

9.13 Surface Integrals

• Surface Area

  

  \[ \begin{aligned} \mathbf{u} &= \Delta x_k \mathbf{i} +f_x(x_k,y_k)\Delta x_k \mathbf{k}\\ \mathbf{v} &= \Delta y_k \mathbf{j} +f_y(x_k,y_k)\Delta y_k \mathbf{k}\\ & \Downarrow \\ \mathbf{u} \times \mathbf{v} &={\scriptsize\left[-f_x(x_k,y_k)\mathbf{i} -f_y(x_k,y_k)\mathbf{j}+\mathbf{k}\right] \Delta x_k \Delta y_k}\\ & \Downarrow \\ \Delta S_k &= {\sqrt{1 +\left[\,f_x(x_k,y_k)\right]^2 +\left[\,f_y(x_k,y_k)\right]^2}\Delta A_k } \end{aligned}\] Then the area of the surface over \(R\) is given by \[ { S=\iint_R \sqrt{1+\left[\,f_x(x,y)\right]^2 +\left[\,f_y(x,y)\right]^2}\,dA} \]

• Surface Integral

  Let \(\,G\,\) be a function of three variables defined over a region of space containing the surface \(S\). \(\,\)Then the surface integral of \(G\) over \(S\) is given by

  \[ \begin{aligned} \lim_{\|P\|\to 0}&\sum_{k=1}^n G(x_k^*,y_k^*,z_k^*)\,\Delta S_k = \iint_S G(x,y,z)\,dS\\ &= \iint_R G(x,y,f(x,y)) \,\sqrt{1+\left[\,f_x(x,y)\right]^2 +\left[\,f_y(x,y)\right]^2}\,dA \end{aligned}\]

• If a smooth surface \(S\) \(\,\)is \(g(x,y,z)=0,\) \(\,\)a unit normal is

  \[\mathbf{n}=\frac{\nabla g}{\|\nabla g\|}\]

  

• Integrals of Vector Fields

  The total volume of a fluid passing through \(\,S\) \(\,\)is called the flux of \(\mathbf{v}\) through \(S\) and is given by

\[ \mathrm{flux} =\iint_S (\mathbf{v}\cdot\mathbf{n})\,dS \]

\(~\)

9.14 Stokes’ Theorem

• Vector Form of Green’s Theorem

  If \(\,\mathbf{f}(x,y)=P(x,y)\mathbf{i} +Q(x,y)\mathbf{j} \,\) is a two-dimensional vector field, \(\text{ }\)then

  \[\mathrm{curl}\, \mathbf{f} =\nabla \times \mathbf{f} =\left|\begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k}\\ \displaystyle\frac{\partial}{\partial x} & \displaystyle\frac{\partial}{\partial y} & \displaystyle\frac{\partial}{\partial z}\\ P & Q & 0 \end{matrix}\right| =\left( \frac{\partial Q}{\partial x} -\frac{\partial P}{\partial y} \right )\mathbf{k}\]

  Green’s theorem can be written in vector notation as

  \[\oint_C \mathbf{f}\cdot \,d\mathbf{r} =\oint_C \mathbf{f}\cdot\mathbf{T}\,ds =\iint_R (\nabla \times \mathbf{f}) \cdot \mathbf{k}\,dA\]

• Stokes’ Theorem - Green’s Theorem in 3-Space

  • Let \(S\) be a piecewise-smooth orientable surface bounded by a piecewise-smooth simple closed curve \(C\)

  • Let \(~\mathbf{f}(x,y,z)=P(x,y,z)\mathbf{i} +Q(x,y,z)\mathbf{j} +R(x,y,z)\mathbf{k} \,\) be a vector field for which \(P\), \(Q\), and \(R\) are continuous and have continuous first partial derivatives in a region of 3-space containing \(S\). \(~\)If \(C\) is traversed in the positive direction, then

  \[\oint_C \mathbf{f}\cdot \,d\mathbf{r} =\oint_C \mathbf{f}\cdot\mathbf{T}\,ds = \iint_R (\nabla \times\mathbf{f}) \cdot \mathbf{n}\,dS\]

• Physical Interpretation of Curl

  \[ \begin{aligned} \oint_{C_r} \mathbf{f} \cdot d\mathbf{r} &= \iint_{S_r} \left[\nabla\times\mathbf{f}(P_0)\right] \cdot \mathbf{n}(P_0)\, dS\\ &= \left[\nabla\times\mathbf{f}(P_0)\right]\, \cdot \mathbf{n}(P_0)\,\iint_{S_r}\,dS =\left[\nabla\times\mathbf{f}(P_0)\right] \cdot \mathbf{n}(P_0)\,S_r\\ &\Downarrow\\ \left[\nabla\times\mathbf{f}(P_0)\right] \cdot \mathbf{n}(P_0) &= \lim_{r\to 0} \frac{1}{S_r}\oint_{C_r} \mathbf{f} \cdot d\mathbf{r} \approx \frac{1}{S_r}\oint_{C_r} \mathbf{f} \cdot d\mathbf{r} \end{aligned}\]

\(~\)

9.15 Triple Integrals

• Let \(F\) be a function of three variables defined over a closed region \(D\) of space. Then the triple integral of \(F\) over \(D\) is given by

  \[{\lim_{\| P\|\to 0} \sum_{k=1}^n F(x_k^*,y_k^*,z_k^*)\,\Delta V_k = \iiint_D F(x,y,z)\,dV}\]

• Evaluation by Iterated Integrals

  \[ \begin{aligned} \iiint_D F(x,y,z)\,dV &= \iint_R \left[ \int_{f_1(x,y)}^{f_2(x,y)} F(x,y,z)\,dz \right]\,dA\\ &= \int_a^b \int_{g_1(x)}^{g_2(x)} \int_{f_1(x,y)}^{f_2(x,y)} F(x,y,z)\,dz\,dy\,dx \end{aligned}\]

  

\(~\)

Example \(\,\) Find the volume of the solid in the first octant bounded by the graphs of \(z=1-y^2\), \(y=2x\), and \(x=3\)

\(~\)

• Cylindrical Coordinates

  \[\begin{aligned} x=r\cos\theta, \;\;&y=r\sin\theta, \;\;z=z \\ &\Downarrow \\ r^2 = x^2 +y^2, \;\; &\tan\theta=\frac{y}{x},\;\; z=z \end{aligned}\]

  

• Triple Integrals in Cylindrical Coordinates

  \[ \begin{aligned} \iiint_D F(r,\theta,z)\,dV &= \iint_R \left[ \int_{f_1(r,\theta)}^{f_2(r,\theta)} F(r,\theta,z)\,dz \right]\,dA \\ &= \int_\alpha^\beta \int_{g_1(\theta)}^{g_2(\theta)} \int_{f_1(r,\theta)}^{f_2(r,\theta)} F(r,\theta,z)\,r\,dz\,dr\,d\theta \end{aligned}\]

  

• Spherical Coordinates

  \[ \begin{aligned} x = \rho\sin\phi\cos\theta, \;\; &y=\rho\sin\phi\sin\theta, \;\; z=\rho\cos\phi \\ &\Downarrow \\ \rho^2 =x^2 +y^2 +z^2, \;\; &\tan\theta=\frac{y}{x}, \;\;\cos\phi=\frac{z}{\sqrt{x^2 +y^2 +z^2}} \end{aligned}\]

• Triple Integrals in Spherical Coordinates

\[ \begin{aligned} \iiint_D &F(\rho,\theta,\phi)\,dV \\ &= \int_\alpha^\beta \int_{g_1(\theta)}^{g_2(\theta)} \int_{f_1(\phi,\theta)}^{f_2(\phi,\theta)} F(\rho,\theta,\phi)\, \rho^2 \sin\phi\, d\rho\,d\phi\,d\theta \end{aligned}\]

9.16 Divergence Theorem

• Another Vector Form of Green’s Theorem

  \[ \begin{aligned} \oint_C (\mathbf{f}\cdot\mathbf{n})\,ds &= \oint_C -Qdx +Pdy \\ &= \iint_R \left[ \frac{\partial P}{\partial x} -\left( -\frac{\partial Q}{\partial y} \right) \right]\,dA = \iint_R \left[ \frac{\partial P}{\partial x} +\frac{\partial Q}{\partial y} \right]\,dA \\ &= \iint_R \nabla\cdot\mathbf{f}\,dA \end{aligned}\]

  where \(\displaystyle\,\mathbf{n}=\frac{dy}{ds}\mathbf{i} -\frac{dx}{ds}\mathbf{j}\)

• Divergence Theorem

  • Let \(D\,\) be a closed and bounded region in 3-space with a piecewise-smooth boundary \(S\) that is oriented outward

  • Let \(\;\mathbf{f}(x,y,z)=P(x,y,z)\,\mathbf{i}+Q(x,y,z)\,\mathbf{j} +R(x,y,z)\,\mathbf{k}~\) be a vector field for which \(P\), \(Q,\) and \(R\) are continuous first partial derivatives in a region of 3-space containing \(D\). \(\,\) Then

  \[\iint_S (\mathbf{f}\cdot \mathbf{n})\,dS =\iiint_D \nabla\cdot\mathbf{f}\,dV\]

• Partial Proof

  

  \[{\scriptsize \displaystyle\iiint_D \frac{\partial R}{\partial z}\,dV =\iint_{R'} \left[ \int_{f_1(x,y)}^{f_2(x,y)} \frac{\partial R}{\partial z}\,dz \right]\,dA =\iint_{R'} \left[ R(x,y,f_2(x,y)) -R(x,y,f_1(x,y)) \right] \,dA }\]

  \[{\scriptsize \displaystyle\iint_S R(\mathbf{k}\cdot\mathbf{n}) \,dS=\iint_{S_1} R(\mathbf{k}\cdot\mathbf{n})\,dS +\iint_{S_2} R(\mathbf{k}\cdot\mathbf{n})\,dS +\iint_{S_3} R(\mathbf{k}\cdot\mathbf{n}) \,dS }\]

  • On \(S_1\): \(\,\) Since the outward normal points downward, \(g(x,y,z)=f_1(x,y)-z=0\). Thus

    \[ \begin{aligned} \mathbf{n} &= \frac{ \frac{\partial f_1}{\partial x} \mathbf{i} +\frac{\partial f_1}{\partial y} \mathbf{j} -\mathbf{k} }{ \sqrt{1+\left(\frac{\partial f_1}{\partial x}\right)^2 +\left(\frac{\partial f_1}{\partial y}\right)^2}} \;\; \Rightarrow \;\; \mathbf{k}\cdot\mathbf{n} =\frac{-1}{\sqrt{1+\left(\frac{\partial f_1}{\partial x}\right)^2 +\left(\frac{\partial f_1}{\partial y}\right)^2}} \\ &\Downarrow \\ \iint_{S_1} &R(\mathbf{k}\cdot\mathbf{n})\,dS = -\iint_{R'} R(x,y,f_1(x,y))\,dA \end{aligned}\]

  • On \(S_2\): \(\,\) Since the outward normal points upward, \(g(x,y,z)=z -f_2(x,y)=0\). Thus

    \[ \begin{aligned} \mathbf{n} &= \frac{ -\frac{\partial f_2}{\partial x} \mathbf{i} -\frac{\partial f_2}{\partial y} \mathbf{j} +\mathbf{k} }{ \sqrt{1+\left(\frac{\partial f_2}{\partial x}\right)^2 +\left(\frac{\partial f_2}{\partial y}\right)^2}} \;\; \Rightarrow \;\; \mathbf{k}\cdot\mathbf{n} =\frac{1}{\sqrt{1+\left(\frac{\partial f_2}{\partial x}\right)^2 +\left(\frac{\partial f_2}{\partial y}\right)^2}} \\ &\Downarrow \\ \iint_{S_2} &R(\mathbf{k}\cdot\mathbf{n})\,dS = \iint_{R'} R(x,y,f_2(x,y))\,dA \end{aligned}\]

  • On \(S_3\): \(\,\) \(\mathbf{k}\cdot\mathbf{n}=0\)

  \[\quad \iint_{S_3} R(\mathbf{k}\cdot\mathbf{n})\,dS = 0\]

• Physical Interpretation of Divergence

  \[ \begin{aligned} \iint_{S_r} (\mathbf{f} \cdot \mathbf{n})\,dS &= \iiint_{D_r} \nabla\cdot\mathbf{f}\,dV \\ &\approx \iiint_{D_r} \nabla\cdot\mathbf{f}(P_0)\,dV = \nabla\cdot\mathbf{f}(P_0) \iiint_{D_r}\,dV = \nabla\cdot\mathbf{f}(P_0) D_r\\ &\Downarrow\\ \nabla\cdot\mathbf{f}(P_0) &= \lim_{r\to 0} \frac{1}{D_r}\iint_{S_r} (\mathbf{f} \cdot \mathbf{n})\,dS \end{aligned}\]

  

• Continuity Equation

  \[ \begin{aligned} \frac{dm}{dt} &= \frac{d}{dt}\iiint_D \rho(x,y,z,t)\,dV = \iiint_D \frac{\partial \rho}{\partial t}\,dV \\ &= -\iint_S (\rho\mathbf{v}\cdot\mathbf{n})\,dS =-\iiint_D \nabla\cdot(\rho\mathbf{v})\,dV\\ &\Downarrow \\ \iiint_D \left[ \frac{\partial \rho}{\partial t} \right. &+\left. \phantom{\frac{}{}}\nabla\cdot(\rho\mathbf{v}) \right]\,dV=0 \\ &\Downarrow \\ \frac{\partial \rho}{\partial t} & +\nabla\cdot(\rho\mathbf{v})=0 \end{aligned}\]

• Euler’s Equation of Motion

  Allowing for the presence of a gravitational body force per unit mass \(\mathbf{g}\), \(\,\)the basic equations of an ideal fluid are

  \[\frac{D \mathbf{v}}{D t} = -\frac{1}{\rho} \nabla p + \mathbf{g}\]

  \[\nabla \cdot \mathbf{v} = 0\]

• Furthermore, it can be helpful to use \(\mathbf{g}=-\nabla \chi\) \(\,\)and the identity

  \[ \left( \mathbf{v} \cdot \nabla \right) \mathbf{v} = \left( \nabla \times \mathbf{v} \right) \times \mathbf{v} + \nabla \left( \frac{1}{2} \mathbf{v}^2 \right)\]

  to cast the momentum equation into the form

  \[ \frac{\partial \mathbf{v}}{\partial t} + \left( \nabla \times \mathbf{v} \right) \times \mathbf{v} = -\nabla \left( \frac{p}{\rho} +\frac{1}{2} \mathbf{v}^2 + \chi \right) =- \nabla H\]

\(~\)

9.17 Change of Variables in Multiple Integrals

• To change the Variable in a Definite Integral

  \[\int_a^b f(x)\,dx =\int_c^d f(g(u))\, g'(u)\,du\]

• Double Integrals

  \[\iint_R F(x,y)\,dA = \iint_R F(f(u,v), g(u,v))\, \left| \frac{\partial(x,y)}{\partial(u,v)} \right|\,dA'\]

  \[\left| \frac{\partial(x,y)}{\partial(u,v)} \right| = \left|\begin{matrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v}\\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{matrix}\right| = h_1 h_2 dq_1 dq_2\]

• Triple Integrals

  \[\iiint_D F(x,y,z)\,dV = \iiint_D F\left(f(u,v,w), g(u,v,w), h(u,v,w)\right)\, \left| \frac{\partial(x,y,z)}{\partial(u,v,w)} \right|\,dV'\]

  \[\left| \frac{\partial (x,y,z)}{\partial (u,v,w)} \right| = \left|\begin{matrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w}\\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w}\\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \end{matrix}\right| = h_1 h_2 h_3 dq_1 dq_2 dq_3\]

\(~\)

Worked Exercises