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method of Lagrange multipliers : a method of solving an optimization problem subject to one or more constraints | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
mixed partial derivatives : second-order or higher partial derivatives, in which at least two of the differentiations are with respect to different variables | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
objective function : the function that is to be maximized or minimized in an optimization problem | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
open set : a setSSthat contains none of its boundary points | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
optimization problem : calculation of a maximum or minimum value of a function of several variables, often using Lagrange multipliers | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
partial derivative : a derivative of a function of more than one independent variable in which all the variables but one are held constant | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
partial differential equation : an equation that involves an unknown function of more than one independent variable and one or more of its partial derivatives | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
region : an open, connected, nonempty subset ofâ2â2 | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
saddle point : given the functionz=f(x,y),z=f(x,y),the point(x0,y0,f(x0,y0))(x0,y0,f(x0,y0))is a saddle point if bothfx(x0,y0)=0fx(x0,y0)=0andfy(x0,y0)=0,fy(x0,y0)=0,butffdoes not have a local extremum at(x0,y0)(x0,y0) | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
surface : the graph of a function of two variables,z=f(x,y)z=f(x,y) | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
tangent plane : given a functionf(x,y)f(x,y)that is differentiable at a point(x0,y0),(x0,y0),the equation of the tangent plane to the surfacez=f(x,y)z=f(x,y)is given byz=f(x0,y0)+fx(x0,y0)(xâx0)+fy(x0,y0)(yây0)z=f(x0,y0)+fx(x0,y0)(xâx0)+fy(x0,y0)(yây0) | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
total differential : the total differential of the functionf(x,y)f(x,y)at(x0,y0)(x0,y0)is given by the formuladz=fx(x0,y0)dx+fy(x0,y0)dydz=fx(x0,y0)dx+fy(x0,y0)dy | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
tree diagram : illustrates and derives formulas for the generalized chain rule, in which each independent variable is accounted for | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
vertical trace : the set of ordered triples(c,y,z)(c,y,z)that solves the equationf(c,y)=zf(c,y)=zfor a given constantx=cx=cor the set of ordered triples(x,d,z)(x,d,z)that solves the equationf(x,d)=zf(x,d)=zfor a given constanty=dy=d | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
δδball : all points inâ3â3lying at a distance of less thanδδfrom(x0,y0,z0)(x0,y0,z0) | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
δδdisk : an open disk of radiusδδcentered at point(a,b)(a,b) | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
We can use a double Riemann sum to approximate the volume of a solid bounded above by a function of two variables over a rectangular region. By taking the limit, this becomes a double integral representing the volume of the solid. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
Properties of double integral are useful to simplify computation and find bounds on their values. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
We can use Fubiniâs theorem to write and evaluate a double integral as an iterated integral. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
Double integrals are used to calculate the area of a region, the volume under a surface, and the average value of a function of two variables over a rectangular region. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
A general bounded regionDDon the plane is a region that can be enclosed inside a rectangular region. We can use this idea to define a double integral over a general bounded region. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
To evaluate an iterated integral of a function over a general nonrectangular region, we sketch the region and express it as a Type I or as a Type II region or as a union of several Type I or Type II regions that overlap only on their boundaries. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
We can use double integrals to find volumes, areas, and average values of a function over general regions, similarly to calculations over rectangular regions. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
We can use Fubiniâs theorem for improper integrals to evaluate some types of improper integrals. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
To apply a double integral to a situation with circular symmetry, it is often convenient to use a double integral in polar coordinates. We can apply these double integrals over a polar rectangular region or a general polar region, using an iterated integral similar to those used with rectangular double integrals. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
The areadAdAin polar coordinates becomesrdrdθ.rdrdθ. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
Usex=rcosθ,x=rcosθ,y=rsinθ,y=rsinθ,anddA=rdrdθdA=rdrdθto convert an integral in rectangular coordinates to an integral in polar coordinates. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
User2=x2+y2r2=x2+y2andθ=tanâ1(yx)θ=tanâ1(yx)to convert an integral in polar coordinates to an integral in rectangular coordinates, if needed. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
To find the volume in polar coordinates bounded above by a surfacez=f(r,θ)z=f(r,θ)over a region on thexyxy-plane, use a double integral in polar coordinates. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
To compute a triple integral we use Fubiniâs theorem, which states that iff(x,y,z)f(x,y,z)is continuous on a rectangular boxB=[a,b]Ã[c,d]Ã[e,f],B=[a,b]Ã[c,d]Ã[e,f],thenâBf(x,y,z)dV=â«efâ«cdâ«abf(x,y,z)dxdydzâBf(x,y,z)dV=â«efâ«cdâ«abf(x,y,z)dxdydzand is also equal to any of the other five possible orderi... | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
To compute the volume of a general solid bounded regionEEwe use the triple integralV(E)=âE1dV.V(E)=âE1dV. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
Interchanging the order of the iterated integrals does not change the answer. As a matter of fact, interchanging the order of integration can help simplify the computation. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
To compute the average value of a function over a general three-dimensional region, we usefave=1V(E)âEf(x,y,z)dV.fave=1V(E)âEf(x,y,z)dV. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
To evaluate a triple integral in cylindrical coordinates, use the iterated integralâ«Î¸=αθ=βâ«r=g1(θ)r=g2(θ)â«z=u1(r,θ)z=u2(r,θ)f(r,θ,z)rdzdrdθ.â«Î¸=αθ=βâ«r=g1(θ)r=g2(θ)â«z=u1(r,θ)z=u2(r,θ)f(r,θ,z)rdzdrdθ. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
To evaluate a triple integral in spherical coordinates, use the iterated integralâ«Î¸=αθ=βâ«Ï=g1(θ)Ï=g2(θ)â«Ï=u1(r,θ)Ï=u2(r,θ)f(Ï,θ,Ï)Ï2sinÏdÏdÏdθ.â«Î¸=αθ=βâ«Ï=g1(θ)Ï=g2(θ)â«Ï=u1(r,θ)Ï=u2(r,θ)f(Ï,θ,Ï)Ï2sinÏdÏdÏdθ. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
Finding the mass, center of mass, moments, and moments of inertia in double integrals: | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
For a laminaRRwith a density functionÏ(x,y)Ï(x,y)at any point(x,y)(x,y)in the plane, the mass ism=â¬RÏ(x,y)dA.m=â¬RÏ(x,y)dA. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
The moments about thex-axisx-axisandy-axisy-axisareMx=â¬RyÏ(x,y)dAandMy=â¬RxÏ(x,y)dA.Mx=â¬RyÏ(x,y)dAandMy=â¬RxÏ(x,y)dA. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
The center of mass is given byxâ=Mym,yâ=Mxm.xâ=Mym,yâ=Mxm. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
The center of mass becomes the centroid of the plane when the density is constant. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
The moments of inertia about thexâaxis,xâaxis,yâaxis,yâaxis,and the origin areIx=â¬Ry2Ï(x,y)dA,Iy=â¬Rx2Ï(x,y)dA,andI0=Ix+Iy=â¬R(x2+y2)Ï(x,y)dA.Ix=â¬Ry2Ï(x,y)dA,Iy=â¬Rx2Ï(x,y)dA,andI0=Ix+Iy=â¬R(x2+y2)Ï(x,y)dA. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
Finding the mass, center of mass, moments, and moments of inertia in triple integrals: | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
For a solid objectQQwith a density functionÏ(x,y,z)Ï(x,y,z)at any point(x,y,z)(x,y,z)in space, the mass ism=âQÏ(x,y,z)dV.m=âQÏ(x,y,z)dV. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
The moments about thexy-plane,xy-plane,thexz-plane,xz-plane,and theyz-planeyz-planeareMxy=âQzÏ(x,y,z)dV,Mxz=âQyÏ(x,y,z)dV,Myz=âQxÏ(x,y,z)dV.Mxy=âQzÏ(x,y,z)dV,Mxz=âQyÏ(x,y,z)dV,Myz=âQxÏ(x,y,z)dV. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
The center of mass is given byxâ=Myzm,yâ=Mxzm,zâ=Mxym.xâ=Myzm,yâ=Mxzm,zâ=Mxym. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
The center of mass becomes the centroid of the solid when the density is constant. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
The moments of inertia about theyz-plane,yz-plane,thexz-plane,xz-plane,and thexy-planexy-planeareIx=âQ(y2+z2)Ï(x,y,z)dV,Iy=âQ(x2+z2)Ï(x,y,z)dV,Iz=âQ(x2+y2)Ï(x,y,z)dV.Ix=âQ(y2+z2)Ï(x,y,z)dV,Iy=âQ(x2+z2)Ï(x,y,z)dV,Iz=âQ(x2+y2)Ï(x,y,z)dV. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
A transformationTTis a function that transforms a regionGGin one plane (space) into a regionRRin another plane (space) by a change of variables. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
A transformationT:GâRT:GâRdefined asT(u,v)=(x,y)T(u,v)=(x,y)(orT(u,v,w)=(x,y,z))(orT(u,v,w)=(x,y,z))is said to be a one-to-one transformation if no two points map to the same image point. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
Ifffis continuous onR,R,thenâ¬Rf(x,y)dA=â¬Sf(g(u,v),h(u,v))|â(x,y)â(u,v)|dudv.â¬Rf(x,y)dA=â¬Sf(g(u,v),h(u,v))|â(x,y)â(u,v)|dudv. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
IfFFis continuous onR,R,thenâRF(x,y,z)dV=âGF(g(u,v,w),h(u,v,w),k(u,v,w))|â(x,y,z)â(u,v,w)|dudvdw=âGH(u,v,w)|J(u,v,w)|dudvdw.âRF(x,y,z)dV=âGF(g(u,v,w),h(u,v,w),k(u,v,w))|â(x,y,z)â(u,v,w)|dudvdw=âGH(u,v,w)|J(u,v,w)|dudvdw. | https://openstax.org/books/calculus-volume-3/pages/5-key-concepts |
⬠R f ( x , y ) d A = lim m , n â â â i = 1 m â j = 1 n f ( x i j * , y i j * ) Î A ⬠R f ( x , y ) d A = lim m , n â â â i = 1 m â j = 1 n f ( x i j * , y i j * ) Î A | https://openstax.org/books/calculus-volume-3/pages/5-key-equations |
â« a b â« c d f ( x , y ) d x d y = â« a b [ â« c d f ( x , y ) d y ] d x â« a b â« c d f ( x , y ) d x d y = â« a b [ â« c d f ( x , y ) d y ] d x or â« c d â« b a f ( x , y ) d x d y = â« c d [ â« a b f ( x , y ) d x ] d y â« c d â« b a f ( x , y ) d x d y = â« c d [ â« a b f ( x , y ) d x ] d y | https://openstax.org/books/calculus-volume-3/pages/5-key-equations |
f ave = 1 Area R ⬠R f ( x , y ) d x d y f ave = 1 Area R ⬠R f ( x , y ) d x d y | https://openstax.org/books/calculus-volume-3/pages/5-key-equations |
⬠D f ( x , y ) d A = ⬠D f ( x , y ) d y d x = ⫠a b [ ⫠g 1 ( x ) g 2 ( x ) f ( x , y ) d y ] d x ⬠D f ( x , y ) d A = ⬠D f ( x , y ) d y d x = ⫠a b [ ⫠g 1 ( x ) g 2 ( x ) f ( x , y ) d y ] d x | https://openstax.org/books/calculus-volume-3/pages/5-key-equations |
⬠D f ( x , y ) d A = ⬠D f ( x , y ) d x d y = ⫠c d [ ⫠h 1 ( y ) h 2 ( y ) f ( x , y ) d x ] d y ⬠D f ( x , y ) d A = ⬠D f ( x , y ) d x d y = ⫠c d [ ⫠h 1 ( y ) h 2 ( y ) f ( x , y ) d x ] d y | https://openstax.org/books/calculus-volume-3/pages/5-key-equations |
⬠R f ( r , θ ) d A = lim m , n â â â i = 1 m â j = 1 n f ( r i j * , θ i j * ) Î A = lim m , n â â â i = 1 m â j = 1 n f ( r i j * , θ i j * ) r i j * Î r Πθ ⬠R f ( r , θ ) d A = lim m , n â â â i = 1 m â j = 1 n f ( r i j * , θ i j * ) Î A = lim m , n â â â i = 1 m â j ... | https://openstax.org/books/calculus-volume-3/pages/5-key-equations |
⬠D f ( r , θ ) r d r d θ = ⫠θ = α θ = β ⫠r = h 1 ( θ ) r = h 2 ( θ ) f ( r , θ ) r d r d θ ⬠D f ( r , θ ) r d r d θ = ⫠θ = α θ = β ⫠r = h 1 ( θ ) r = h 2 ( θ ) f ( r , θ ) r d r d θ | https://openstax.org/books/calculus-volume-3/pages/5-key-equations |
lim l , m , n â â â i = 1 l â j = 1 m â k = 1 n f ( x i j k * , y i j k * , z i j k * ) Î x Î y Î z = â B f ( x , y , z ) d V lim l , m , n â â â i = 1 l â j = 1 m â k = 1 n f ( x i j k * , y i j k * , z i j k * ) Î x Î y Î z = â B f ( x , y , z ) d V | https://openstax.org/books/calculus-volume-3/pages/5-key-equations |
â B g ( x , y , z ) d V = â B g ( r cos θ , r sin θ , z ) r d r d θ d z = â B f ( r , θ , z ) r d r d θ d z â B g ( x , y , z ) d V = â B g ( r cos θ , r sin θ , z ) r d r d θ d z = â B f ( r , θ , z ) r d r d θ d z | https://openstax.org/books/calculus-volume-3/pages/5-key-equations |
â B f ( Ï , θ , Ï ) Ï 2 sin Ï d Ï d Ï d θ = â« Ï = γ Ï = Ï â« Î¸ = α θ = β â« Ï = a Ï = b f ( Ï , θ , Ï ) Ï 2 sin Ï d Ï d Ï d θ â B f ( Ï , θ , Ï ) Ï 2 sin Ï d Ï d Ï d θ = â« Ï = γ Ï = Ï â« Î¸ = α θ = β â« Ï = a Ï = b f ( Ï , θ , Ï ) Ï 2 sin Ï d Ï d Ï d θ | https://openstax.org/books/calculus-volume-3/pages/5-key-equations |
m = lim k , l â â â i = 1 k â j = 1 l m i j = lim k , l â â â i = 1 k â j = 1 l Ï ( x i j * , y i j * ) Î A = ⬠R Ï ( x , y ) d A m = lim k , l â â â i = 1 k â j = 1 l m i j = lim k , l â â â i = 1 k â j = 1 l Ï ( x i j * , y i j * ) Î A = ⬠R Ï ( x , y ) d A | https://openstax.org/books/calculus-volume-3/pages/5-key-equations |
M x = lim k , l â â â i = 1 k â j = 1 l ( y i j * ) m i j = lim k , l â â â i = 1 k â j = 1 l ( y i j * ) Ï ( x i j * , y i j * ) Î A = ⬠R y Ï ( x , y ) d A M x = lim k , l â â â i = 1 k â j = 1 l ( y i j * ) m i j = lim k , l â â â i = 1 k â j = 1 l ( y i j * ) Ï ( x i j * , y i ... | https://openstax.org/books/calculus-volume-3/pages/5-key-equations |
M y = lim k , l â â â i = 1 k â j = 1 l ( x i j * ) m i j = lim k , l â â â i = 1 k â j = 1 l ( x i j * ) Ï ( x i j * , y i j * ) Î A = ⬠R x Ï ( x , y ) d A M y = lim k , l â â â i = 1 k â j = 1 l ( x i j * ) m i j = lim k , l â â â i = 1 k â j = 1 l ( x i j * ) Ï ( x i j * , y i ... | https://openstax.org/books/calculus-volume-3/pages/5-key-equations |
x â = M y m = ⬠R x Ï ( x , y ) d A ⬠R Ï ( x , y ) d A x â = M y m = ⬠R x Ï ( x , y ) d A ⬠R Ï ( x , y ) d A and y â = M x m = ⬠R y Ï ( x , y ) d A ⬠R Ï ( x , y ) d A y â = M x m = ⬠R y Ï ( x , y ) d A ⬠R Ï ( x , y ) d A | https://openstax.org/books/calculus-volume-3/pages/5-key-equations |
double integral : of the functionf(x,y)f(x,y)over the regionRRin thexyxy-plane is defined as the limit of a double Riemann sum,â¬Rf(x,y)dA=limm,nâââi=1mâj=1nf(xij*,yij*)ÎA.â¬Rf(x,y)dA=limm,nâââi=1mâj=1nf(xij*,yij*)ÎA. | https://openstax.org/books/calculus-volume-3/pages/5-key-terms |
double Riemann sum : of the functionf(x,y)f(x,y)over a rectangular regionRRisâi=1mâj=1nf(xij*,yij*)ÎAâi=1mâj=1nf(xij*,yij*)ÎAwhereRRis divided into smaller subrectanglesRijRijand(xij*,yij*)(xij*,yij*)is an arbitrary point inRijRij | https://openstax.org/books/calculus-volume-3/pages/5-key-terms |
Fubiniâs theorem : iff(x,y)f(x,y)is a function of two variables that is continuous over a rectangular regionR={(x,y)ââ2|aâ¤xâ¤b,câ¤yâ¤d},R={(x,y)ââ2|aâ¤xâ¤b,câ¤yâ¤d},then the double integral offfover the region equals an iterated integral,â¬Rf(x,y)dydx=â«abâ«cdf(x,y)dxdy=â«cdâ«abf(x,y)dxdyâ¬Rf(x... | https://openstax.org/books/calculus-volume-3/pages/5-key-terms |
improper double integral : a double integral over an unbounded region or of an unbounded function | https://openstax.org/books/calculus-volume-3/pages/5-key-terms |
iterated integral : for a functionf(x,y)f(x,y)over the regionRRisâ«abâ«cdf(x,y)dxdy=â«ab[â«cdf(x,y)dy]dx,â«abâ«cdf(x,y)dxdy=â«ab[â«cdf(x,y)dy]dx,â«cdâ«baf(x,y)dxdy=â«cd[â«abf(x,y)dx]dy,â«cdâ«baf(x,y)dxdy=â«cd[â«abf(x,y)dx]dy,wherea,b,c,a,b,c,andddare any real numbers andR=[a,b]Ã[c,d]R=[a,b]Ã[c,d] | https://openstax.org/books/calculus-volume-3/pages/5-key-terms |
Jacobian : the JacobianJ(u,v)J(u,v)in two variables is a2Ã22Ã2determinant:J(u,v)=|âxâuâyâuâxâvâyâv|;J(u,v)=|âxâuâyâuâxâvâyâv|;the JacobianJ(u,v,w)J(u,v,w)in three variables is a3Ã33Ã3determinant:J(u,v,w)=|âxâuâyâuâzâuâxâvâyâvâzâvâxâwâyâwâzâw|J(u,v,... | https://openstax.org/books/calculus-volume-3/pages/5-key-terms |
one-to-one transformation : a transformationT:GâRT:GâRdefined asT(u,v)=(x,y)T(u,v)=(x,y)is said to be one-to-one if no two points map to the same image point | https://openstax.org/books/calculus-volume-3/pages/5-key-terms |
planar transformation : a functionTTthat transforms a regionGGin one plane into a regionRRin another plane by a change of variables | https://openstax.org/books/calculus-volume-3/pages/5-key-terms |
polar rectangle : the region enclosed between the circlesr=ar=aandr=br=band the anglesθ=αθ=αandθ=β;θ=β;it is described asR={(r,θ)|aâ¤râ¤b,αâ¤Î¸â¤Î²}R={(r,θ)|aâ¤râ¤b,αâ¤Î¸â¤Î²} | https://openstax.org/books/calculus-volume-3/pages/5-key-terms |
radius of gyration : the distance between the rotational axis of the object and the point where the entire mass of the object can be concentrated and have the same moment of inertia | https://openstax.org/books/calculus-volume-3/pages/5-key-terms |
transformation : a function that transforms a regionGGin one plane into a regionRRin another plane by a change of variables | https://openstax.org/books/calculus-volume-3/pages/5-key-terms |
triple integral : the triple integral of a continuous functionf(x,y,z)f(x,y,z)over a rectangular solid boxBBis the limit of a Riemann sum for a function of three variables, if this limit exists | https://openstax.org/books/calculus-volume-3/pages/5-key-terms |
triple integral in cylindrical coordinates : the limit of a triple Riemann sum, provided the following limit exists:liml,m,nâââi=1lâj=1mâk=1nf(rijk*,θijk*,zijk*)rijk*ÎrÎθÎzliml,m,nâââi=1lâj=1mâk=1nf(rijk*,θijk*,zijk*)rijk*ÎrÎθÎz | https://openstax.org/books/calculus-volume-3/pages/5-key-terms |
triple integral in spherical coordinates : the limit of a triple Riemann sum, provided the following limit exists:liml,m,nâââi=1lâj=1mâk=1nf(Ïijk*,θijk*,Ïijk*)(Ïijk*)2sinÏÎÏÎθÎÏliml,m,nâââi=1lâj=1mâk=1nf(Ïijk*,θijk*,Ïijk*)(Ïijk*)2sinÏÎÏÎθÎÏ | https://openstax.org/books/calculus-volume-3/pages/5-key-terms |
Type I : a regionDDin thexyxy-plane is Type I if it lies between two vertical lines and the graphs of two continuous functionsg1(x)g1(x)andg2(x)g2(x) | https://openstax.org/books/calculus-volume-3/pages/5-key-terms |
Type II : a regionDDin thexyxy-plane is Type II if it lies between two horizontal lines and the graphs of two continuous functionsh1(y)andh2(y)h1(y)andh2(y) | https://openstax.org/books/calculus-volume-3/pages/5-key-terms |
A vector field assigns a vectorF(x,y)F(x,y)to each point(x,y)(x,y)in a subsetDofâ2orâ3.â2orâ3.F(x,y,z)F(x,y,z)to each point(x,y,z)(x,y,z)in a subsetDofâ3.â3. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
Vector fields can describe the distribution of vector quantities such as forces or velocities over a region of the plane or of space. They are in common use in such areas as physics, engineering, meteorology, oceanography. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
We can sketch a vector field by examining its defining equation to determine relative magnitudes in various locations and then drawing enough vectors to determine a pattern. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
A vector fieldFFis called conservative if there exists a scalar functionffsuch thatâf=F.âf=F. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
Line integrals generalize the notion of a single-variable integral to higher dimensions. The domain of integration in a single-variable integral is a line segment along thex-axis, but the domain of integration in a line integral is a curve in a plane or in space. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
IfCis a curve, then the length ofCisâ«Cds.â«Cds. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
There are two kinds of line integral: scalar line integrals and vector line integrals. Scalar line integrals can be used to calculate the mass of a wire; vector line integrals can be used to calculate the work done on a particle traveling through a field. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
Scalar line integrals can be calculated usingEquation 6.8; vector line integrals can be calculated usingEquation 6.9. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
Two key concepts expressed in terms of line integrals are flux and circulation. Flux measures the rate that a field crosses a given line; circulation measures the tendency of a field to move in the same direction as a given closed curve. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
The theorems in this section require curves that are closed, simple, or both, and regions that are connected or simply connected. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
The line integral of a conservative vector field can be calculated using the Fundamental Theorem for Line Integrals. This theorem is a generalization of the Fundamental Theorem of Calculus in higher dimensions. Using this theorem usually makes the calculation of the line integral easier. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
Conservative fields are independent of path. The line integral of a conservative field depends only on the value of the potential function at the endpoints of the domain curve. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
Given vector fieldF, we can test whetherFis conservative by using the cross-partial property. IfFhas the cross-partial property and the domain is simply connected, thenFis conservative (and thus has a potential function). IfFis conservative, we can find a potential function by using the Problem-Solving Strategy. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
The circulation of a conservative vector field on a simply connected domain over a closed curve is zero. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
Greenâs theorem relates the integral over a connected region to an integral over the boundary of the region. Greenâs theorem is a version of the Fundamental Theorem of Calculus in one higher dimension. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
Greenâs Theorem comes in two forms: a circulation form and a flux form. In the circulation form, the integrand isF·T.F·T.In the flux form, the integrand isF·N.F·N. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
Greenâs theorem can be used to transform a difficult line integral into an easier double integral, or to transform a difficult double integral into an easier line integral. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
A vector field is source free if it has a stream function. The flux of a source-free vector field across a closed curve is zero, just as the circulation of a conservative vector field across a closed curve is zero. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
The divergence of a vector field is a scalar function. Divergence measures the âoutflowing-nessâ of a vector field. Ifvis the velocity field of a fluid, then the divergence ofvat a point is the outflow of the fluid less the inflow at the point. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
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