Vector Calculus

Chapter Review

THINGS TO KNOW

15.1 Vector Fields

A vector field \(\mathbf{F}\) in the plane: \(\mathbf{F}=\mathbf{F}(x,y)=P(x,y)\mathbf{i}+Q(x,y)\mathbf{j}\) (p. 974)
A vector field \(\mathbf{F}\) in space: \(\mathbf{F}=\mathbf{F}(x,y,z)=P(x,y,z)\mathbf{i}+Q(x,y,z)\mathbf{j}+R(x,y,z)\mathbf{k}\) (p. 974)

15.2 Line Integrals

Line integral of \(f\) along a smooth curve \(C\): \(\int_{C}f(x,y)\,ds= \int_{a}^{b}f(x(t),y(t))\sqrt{\left( \dfrac{dx}{dt}\right) ^{2}+\left( \dfrac{dy}{dt}\right) ^{2}}\,dt\) (p. 978)
The line integral of \(P\,dx+Q\,dy\) along a smooth curve \(C\) in the plane:
\(\int_{C}\mathbf{F}\,{\cdot}\, d\mathbf{r}=\int_{C}(P\,dx+Q\,dy)\) (p. 983)
The line integral of \(P\,dx+Q\,dy+Rdz\) along a smooth curve \(C\) in space: \(\int_{C}\mathbf{F}\,{\cdot}\, d\mathbf{r} =\int_{C}(P\,dx+Qdy+R\,dz)\) (p. 986)
A piecewise-smooth curve \(C\) consists of a finite number of smooth curves that are joined together end to end. (p. 984)

15.3 Fundamental Theorem of Line Integrals

\(\mathbf{F}\) is a conservative vector field if \(\mathbf{F}=\boldsymbol {\nabla}\ f\); the function \(f\) is called the potential function for \(\mathbf{F}\). (p. 990)
A vector field \(\mathbf{F}\) is conservative if and only if \(\dfrac{ \partial P}{\partial y}=\dfrac{\partial Q}{\partial x}\).
(p. 997)
A piecewise-smooth curve \(C\) is closed if its initial point and terminal point coincide. The curve \(C\) is simple if it does not intersect itself. (p. 996)
Simply connected region (p. 996)
Fundamental Theorem of Line Integrals: If \(\mathbf{F}\) is a conservative field, then \(\int_{C}\mathbf{F}\,{\cdot}\, d\mathbf{r}=\int_{(x_{0},y_{0})}^{(x_{1},y_{1})}\mathbf{F}\,{\cdot}\, d\mathbf{r}=f(x_{1},y_{1})-f(x_{0},y_{0})\). (p. 990)
If \(\mathbf{F}\) is a conservative vector field on an open region \(R\) and \(C\) is a closed, piecewise-smooth curve in \(R,\) then \(\int_{C}\mathbf{F}\,{\cdot}\, d\mathbf{r}=\int_{C}(P\,dx+Q\,dy)=0.\) (p. 992)

1054

15.4 An Application of Line Integrals: Work

Work: \(W=\int_{C}\mathbf{F}\,{\cdot}\, d\mathbf{r}=\int_{C}(P\,dx+Q\,dy)\) (p. 1003)
If \(\mathbf{F}\) is a conservative force field, the work \(W\) done by \(\mathbf{F}\) in moving an object along a closed path is 0. (p. 1004)
Law of Conservation of Energy (p. 1006)

15.5 Green's Theorem

Green's Theorem:
\(\oint_{\!\!\!C}(P\,dx+Q\,dy)=\iint\limits_{\kern-3ptR}\left( \dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y}\right) dx\,dy \) (p. 1008)
Green's Theorem for Area: \(A=\dfrac{1}{2}\oint_{\!\!\!C}(-y\,dx+x\,dy)\)
(p. 1010)

15.6 Parametric Surfaces

Parametrization; parameter domain (p. 1016)
Coordinate curves (p. 1017)
Smooth parametrization; smooth surface (p. 1021)
Surface area of a parametrized surface: \(S=\iint\limits_{\kern-3ptR}dS=\iint \limits_{\kern-8ptR}\left\Vert \mathbf{r}_{u}\times \mathbf{r}_{v}\right\Vert \,du\,dv \) (p. 1024)

15.7 Surface and Flux Integrals

A surface integral expressed as a double integral: \(\iint \limits_{\kern-8ptS}F(x,y,z)\,dS=\iint\limits_{\kern-3ptR}F(x(u,v) ,y(u,v) ,\,z(u,v))\left\Vert \mathbf{r}_{u}\times \mathbf{r} _{v}\right\Vert \,du\,dv\) (p. 1027)
Orientable surface (p. 1030)
The flux of \(\mathbf{F}\) across \(S: \dfrac{\hbox{Mass of fluid}}{ \hbox{Unit time}}=\iint\limits_{\kern-3ptS}\rho \mathbf{F}\,{\cdot}\, \mathbf{n}\,dS,\) where \(\rho \) is the mass density of the fluid, the vector field \(\mathbf{F}\) is the velocity of the fluid, and \(\mathbf{n}\) is a unit normal vector of \( S \) (p. 1034)

15.8 The Divergence Theorem

\({div}\mathbf{F}=\dfrac{\partial P}{\partial x}+\dfrac{\partial Q }{\partial y}+\) \(\dfrac{\partial R}{\partial z}\) (p. 1038)
Divergence Theorem: \(\ \iiint\limits_{\kern-13ptE}{div}\mathbf{F} \,dV=\iint\limits_{\kern-3ptS}\mathbf{F}\,{\cdot}\, \mathbf{n}\,dS\) (p. 1038)

15.9 Stokes' Theorem

\({curl}\mathbf{F}=\left( \dfrac{\partial R}{\partial y}-\dfrac{ \partial Q}{\partial z}\right) \,\mathbf{i}-\left( \dfrac{\partial R}{ \partial x}-\dfrac{\partial P}{\partial z}\right) \,\mathbf{j}+\left( \dfrac{ \partial Q}{\partial x}-\dfrac{\partial P}{\partial y}\right) \mathbf{k}\) (p. 1046)
Stokes' Theorem: \(\oint_{C}\mathbf{F}\,{\cdot}\, d\mathbf{r} =\iint\limits_{\kern-3ptS}{curl}\mathbf{F}\,{\cdot}\, \mathbf{n}\,dS\) (p. 1047)
\(\mathbf{F}\) is a conservative vector field in space if and only if \( {curl}\mathbf{F}=0\) (p. 1049)

OBJECTIVES

Section	You should be able to...	Examples	Review Exercises
15.1	1 Describe a vector field (p. 974)	1–6	1, 9
15.2	1 Define a line integral in the plane (p. 977)
	2 Find the value of a line integral along a smooth curve (p. 979)	1–4	11, 15
	3 Find line integrals of the form \(\int_{C}f(x,y) dx\) and \(\int_{C}f(x,y) dy\) (p. 981)	5–7	3, 4, 12(a),(c), 14, 23
	4 Find line integrals along a piecewise-smooth curve (p. 984)	8	12(b), 21
	5 Find the value of a line integral in space (p. 985)	9	7, 8
15.3	1 Identify a conservative vector field and a potential function (p. 990)	1	9(a)
	2 Use the Fundamental Theorem of Line Integrals (p. 991)	2, 3	6(b), 9(b),10
	3 Reconstruct a function from its gradient: Finding the potential function for a conservative vector field (p. 995)	4	6(a)
	4 Determine whether a vector field is conservative (p. 996)	5–8	5
15.4	1 Compute work (p. 1003)	1, 2	16–18
15.5	1 Use Green's Theorem to find a line integral (p. 1009)	1, 2	20, 22, 24
	2 Use Green's Theorem to find area (p. 1010)	3	19
	3 Use Green's Theorem with multiply-connected regions (p. 1011)	4	25
15.6	1 Describe surfaces defined parametrically (p. 1017)	1, 2	26
	2 Find a parametric representation of a surface (p. 1018)	3–5	27
	3 Find equations for a tangent plane and a normal line (p. 1022)	6, 7	28
	4 Find the surface area of a parametrized surface (p. 1023)	8	29
15.7	1 Find a surface integral using a double integral (p. 1027)	1–3	13, 30, 31, 33–35
	2 Determine the orientation of a surface (p. 1030)	4–6	32
	3 Find the flux of a vector field across a surface (p. 1033)	7, 8	36, 46(a)
15.8	1 Find the divergence of a vector field (p. 1038)	1	37(a)–40(a)
	2 Use the Divergence Theorem (p. 1038)	2–4	41, 43, 44
	3 Interpret the divergence of \(\mathbf{F}\) (p. 1042)	5	46(a)
15.9	1 Find the curl of \(\mathbf{F}\) (p. 1046)	1	37(b)–40(b)
	2 Verify Stokes' Theorem (p. 1047)	2	37(c)–40(c)
	3 Use Stokes' Theorem to find an integral (p. 1048)	3, 4	42, 45
	4 Use Stokes' Theorem with conservative vector fields (p. 1049)	5	46(b), 47, 48
	5 Interpret the curl of \(\mathbf{F}\) (p. 1050)