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The curl of a vector field is a vector field. The curl of a vector field at pointPmeasures the tendency of particles atPto rotate about the axis that points in the direction of the curl atP. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
A vector field with a simply connected domain is conservative if and only if its curl is zero. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
Surfaces can be parameterized, just as curves can be parameterized. In general, surfaces must be parameterized with two parameters. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
Surfaces can sometimes be oriented, just as curves can be oriented. Some surfaces, such as a Möbius strip, cannot be oriented. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
A surface integral is like a line integral in one higher dimension. The domain of integration of a surface integral is a surface in a plane or space, rather than a curve in a plane or space. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
The integrand of a surface integral can be a scalar function or a vector field. To calculate a surface integral with an integrand that is a function, useEquation 6.19. To calculate a surface integral with an integrand that is a vector field, useEquation 6.20. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
IfSis a surface, then the area ofSisâ«â«SdS.â«â«SdS. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
Stokesâ theorem relates a flux integral over a surface to a line integral around the boundary of the surface. Stokesâ theorem is a higher dimensional version of Greenâs theorem, and therefore is another version of the Fundamental Theorem of Calculus in higher dimensions. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
Stokesâ theorem can be used to transform a difficult surface integral into an easier line integral, or a difficult line integral into an easier surface integral. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
Through Stokesâ theorem, line integrals can be evaluated using the simplest surface with boundaryC. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
Faradayâs law relates the curl of an electric field to the rate of change of the corresponding magnetic field. Stokesâ theorem can be used to derive Faradayâs law. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
The divergence theorem relates a surface integral across closed surfaceSto a triple integral over the solid enclosed byS. The divergence theorem is a higher dimensional version of the flux form of Greenâs theorem, and is therefore a higher dimensional version of the Fundamental Theorem of Calculus. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
The divergence theorem can be used to transform a difficult flux integral into an easier triple integral and vice versa. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
The divergence theorem can be used to derive Gaussâ law, a fundamental law in electrostatics. | https://openstax.org/books/calculus-volume-3/pages/6-key-concepts |
F ( x , y ) = ⩠P ( x , y ) , Q ( x , y ) ⪠F ( x , y ) = ⩠P ( x , y ) , Q ( x , y ) ⪠or F ( x , y ) = P ( x , y ) i + Q ( x , y ) j F ( x , y ) = P ( x , y ) i + Q ( x , y ) j | https://openstax.org/books/calculus-volume-3/pages/6-key-equations |
F ( x , y , z ) = ⩠P ( x , y , z ) , Q ( x , y , z ) , R ( x , y , z ) ⪠F ( x , y , z ) = ⩠P ( x , y , z ) , Q ( x , y , z ) , R ( x , y , z ) ⪠or F ( x , y , z ) = P ( x , y , z ) i + Q ( x , y , z ) j + R ( x , y , z ) k F ( x , y , z ) = P ( x , y , z ) i + Q ( x , y , z ) j + R ( x , y , z ) k | https://openstax.org/books/calculus-volume-3/pages/6-key-equations |
â« C f ( x , y , z ) d s = â« a b f ( r ( t ) ) ( x â² ( t ) ) 2 + ( y â² ( t ) ) 2 + ( z â² ( t ) ) 2 d t â« C f ( x , y , z ) d s = â« a b f ( r ( t ) ) ( x â² ( t ) ) 2 + ( y â² ( t ) ) 2 + ( z â² ( t ) ) 2 d t | https://openstax.org/books/calculus-volume-3/pages/6-key-equations |
⫠C F · d r = ⫠C F · T d s = ⫠a b F ( r ( t ) ) · r Ⲡ( t ) d t ⫠C F · d r = ⫠C F · T d s = ⫠a b F ( r ( t ) ) · r Ⲡ( t ) d t or ⫠C P d x + Q d y + R d z = ⫠a b ( P ( r ( t ) ) d x d t + Q ( r ( t ) ) d y d t + R ( r ( t ) ) d z d t ) d t ⫠C P d x + Q d y + R d z = ⫠a b ( P ( r (... | https://openstax.org/books/calculus-volume-3/pages/6-key-equations |
â« C F · n ( t ) â n ( t ) â d s = â« a b F ( r ( t ) ) · n ( t ) d t â« C F · n ( t ) â n ( t ) â d s = â« a b F ( r ( t ) ) · n ( t ) d t | https://openstax.org/books/calculus-volume-3/pages/6-key-equations |
â« C â f · d r = f ( r ( b ) ) â f ( r ( a ) ) â« C â f · d r = f ( r ( b ) ) â f ( r ( a ) ) | https://openstax.org/books/calculus-volume-3/pages/6-key-equations |
â« C â f · d r = 0 â« C â f · d r = 0 | https://openstax.org/books/calculus-volume-3/pages/6-key-equations |
â« C P d x + Q d y = ⬠D Q x â P y d A , â« C P d x + Q d y = ⬠D Q x â P y d A , where C is the boundary of D | https://openstax.org/books/calculus-volume-3/pages/6-key-equations |
⫠C F · N d s = ⬠D P x + Q y d A ⫠C F · N d s = ⬠D P x + Q y d A | https://openstax.org/books/calculus-volume-3/pages/6-key-equations |
â« â D F · d r = ⬠D Q x â P y d A â« â D F · d r = ⬠D Q x â P y d A | https://openstax.org/books/calculus-volume-3/pages/6-key-equations |
â Ã F = ( R y â Q z ) i + ( P z â R x ) j + ( Q x â P y ) k â Ã F = ( R y â Q z ) i + ( P z â R x ) j + ( Q x â P y ) k | https://openstax.org/books/calculus-volume-3/pages/6-key-equations |
â · F = P x + Q y + R z â · F = P x + Q y + R z | https://openstax.org/books/calculus-volume-3/pages/6-key-equations |
â · ( â à F ) = 0 â · ( â à F ) = 0 | https://openstax.org/books/calculus-volume-3/pages/6-key-equations |
â Ã ( â f ) = 0 â Ã ( â f ) = 0 | https://openstax.org/books/calculus-volume-3/pages/6-key-equations |
⫠⫠S f ( x , y , z ) d S = ⫠⫠D f ( r ( u , v ) ) | | t u à t v | | d A ⫠⫠S f ( x , y , z ) d S = ⫠⫠D f ( r ( u , v ) ) | | t u à t v | | d A | https://openstax.org/books/calculus-volume-3/pages/6-key-equations |
⬠S F · N d S = ⬠S F · d S = ⬠D F ( r ( u , v ) ) · ( t u à t v ) d A ⬠S F · N d S = ⬠S F · d S = ⬠D F ( r ( u , v ) ) · ( t u à t v ) d A | https://openstax.org/books/calculus-volume-3/pages/6-key-equations |
⫠C F · d r = ⬠S curl F · d S ⫠C F · d r = ⬠S curl F · d S | https://openstax.org/books/calculus-volume-3/pages/6-key-equations |
â E div F d V = ⬠S F · d S â E div F d V = ⬠S F · d S | https://openstax.org/books/calculus-volume-3/pages/6-key-equations |
circulation : the tendency of a fluid to move in the direction of curveC. IfCis a closed curve, then the circulation ofFalongCis line integralâ«CF·Tds,â«CF·Tds,which we also denoteâ«CF·Tdsâ«CF·Tds | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
closed curve : a curve for which there exists a parameterizationr(t),r(t),aâ¤tâ¤b,aâ¤tâ¤b,such thatr(a)=r(b),r(a)=r(b),and the curve is traversed exactly once | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
closed curve : a curve that begins and ends at the same point | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
connected region : a region in which any two points can be connected by a path with a trace contained entirely inside the region | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
conservative field : a vector field for which there exists a scalar functionffsuch thatâf=Fâf=F | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
curl : the curl of vector fieldF=â©P,Q,Râª,F=â©P,Q,Râª,denotedâÃF,âÃF,is the âdeterminantâ of the matrix|ijkââxââyââzPQR||ijkââxââyââzPQR|and is given by the expression(RyâQz)i+(PzâRx)j+(QxâPy)k;(RyâQz)i+(PzâRx)j+(QxâPy)k;it measures the tendency of particles at a point... | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
divergence : the divergence of a vector fieldF=â©P,Q,Râª,F=â©P,Q,Râª,denotedââF,ââF,isPx+Qy+Rz;Px+Qy+Rz;it measures the âoutflowing-nessâ of a vector field | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
divergence theorem : a theorem used to transform a difficult flux integral into an easier triple integral and vice versa | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
flux : the rate of a fluid flowing across a curve in a vector field; the flux of vector fieldFacross plane curveCis line integralâ«CF·n(t)ân(t)âdsâ«CF·n(t)ân(t)âds | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
flux integral : another name for a surface integral of a vector field; the preferred term in physics and engineering | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
Fundamental Theorem for Line Integrals : the value of line integralâ«Câf·drâ«Câf·drdepends only on the value offfat the endpoints ofC:â«Câf·dr=f(r(b))âf(r(a))â«Câf·dr=f(r(b))âf(r(a)) | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
Gaussâ law : ifSis a piecewise, smooth closed surface in a vacuum andQis the total stationary charge inside ofS, then the flux of electrostatic fieldEacrossSisQ/ε0Q/ε0 | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
gradient field : a vector fieldFFfor which there exists a scalar functionffsuch thatâf=F;âf=F;in other words, a vector field that is the gradient of a function; such vector fields are also calledconservative | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
Greenâs theorem : relates the integral over a connected region to an integral over the boundary of the region | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
grid curves : curves on a surface that are parallel to grid lines in a coordinate plane | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
heat flow : a vector field proportional to the negative temperature gradient in an object | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
independence of path : a vector fieldFhas path independence ifâ«C1F·dr=â«C2F·drâ«C1F·dr=â«C2F·drfor any curvesC1C1andC2C2in the domain ofFwith the same initial points and terminal points | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
inverse-square law : the electrostatic force at a given point is inversely proportional to the square of the distance from the source of the charge | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
line integral : the integral of a function along a curve in a plane or in space | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
mass flux : the rate of mass flow of a fluid per unit area, measured in mass per unit time per unit area | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
orientation of a curve : the orientation of a curveCis a specified direction ofC | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
orientation of a surface : if a surface has an âinnerâ side and an âouterâ side, then an orientation is a choice of the inner or the outer side; the surface could also have âupwardâ and âdownwardâ orientations | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
parameter domain (parameter space) : the region of theuvplane over which the parametersuandvvary for parameterizationr(u,v)=â©x(u,v),y(u,v),z(u,v)âªr(u,v)=â©x(u,v),y(u,v),z(u,v)⪠| https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
parameterized surface (parametric surface) : a surface given by a description of the formr(u,v)=â©x(u,v),y(u,v),z(u,v)âª,r(u,v)=â©x(u,v),y(u,v),z(u,v)âª,where the parametersuandvvary over a parameter domain in theuv-plane | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
piecewise smooth curve : an oriented curve that is not smooth, but can be written as the union of finitely many smooth curves | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
potential function : a scalar functionffsuch thatâf=Fâf=F | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
radial field : a vector field in which all vectors either point directly toward or directly away from the origin; the magnitude of any vector depends only on its distance from the origin | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
regular parameterization : parameterizationr(u,v)=â©x(u,v),y(u,v),z(u,v)âªr(u,v)=â©x(u,v),y(u,v),z(u,v)âªsuch thatruÃrvruÃrvis not zero for any point(u,v)(u,v)in the parameter domain | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
rotational field : a vector field in which the vector at point(x,y)(x,y)is tangent to a circle with radiusr=x2+y2;r=x2+y2;in a rotational field, all vectors flow either clockwise or counterclockwise, and the magnitude of a vector depends only on its distance from the origin | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
scalar line integral : the scalar line integral of a functionffalong a curveCwith respect to arc length is the integralâ«Cfds,â«Cfds,it is the integral of a scalar functionffalong a curve in a plane or in space; such an integral is defined in terms of a Riemann sum, as is a single-variable integral | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
simple curve : a curve that does not cross itself | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
simply connected region : a region that is connected and has the property that any closed curve that lies entirely inside the region encompasses points that are entirely inside the region | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
Stokesâ theorem : relates the flux integral over a surfaceSto a line integral around the boundaryCof the surfaceS | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
stream function : ifF=â©P,QâªF=â©P,Qâªis a source-free vector field, then stream functiongis a function such thatP=gyP=gyandQ=âgxQ=âgx | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
surface area : the area of surfaceSgiven by the surface integralâ«â«SdSâ«â«SdS | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
surface independent : flux integrals of curl vector fields are surface independent if their evaluation does not depend on the surface but only on the boundary of the surface | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
surface integral : an integral of a function over a surface | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
surface integral of a scalar-valued function : a surface integral in which the integrand is a scalar function | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
surface integral of a vector field : a surface integral in which the integrand is a vector field | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
unit vector field : a vector field in which the magnitude of every vector is 1 | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
vector field : measured inâ2,â2,an assignment of a vectorF(x,y)F(x,y)to each point(x,y)(x,y)of a subsetDDofâ2;â2;inâ3,â3,an assignment of a vectorF(x,y,z)F(x,y,z)to each point(x,y,z)(x,y,z)of a subsetDDofâ3â3 | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
vector line integral : the vector line integral of vector fieldFalong curveCis the integral of the dot product ofFwith unit tangent vectorTofCwith respect to arc length,â«CF·Tds;â«CF·Tds;such an integral is defined in terms of a Riemann sum, similar to a single-variable integral | https://openstax.org/books/calculus-volume-3/pages/6-key-terms |
Second-order differential equations can be classified as linear or nonlinear, homogeneous or nonhomogeneous. | https://openstax.org/books/calculus-volume-3/pages/7-key-concepts |
To find a general solution for a homogeneous second-order differential equation, we must find two linearly independent solutions. Ify1(x)y1(x)andy2(x)y2(x)are linearly independent solutions to a second-order, linear, homogeneous differential equation, then the general solution is given byy(x)=c1y1(x)+c2y2(x).y(x)=c1y1(... | https://openstax.org/books/calculus-volume-3/pages/7-key-concepts |
To solve homogeneous second-order differential equations with constant coefficients, find the roots of the characteristic equation. The form of the general solution varies depending on whether the characteristic equation has distinct, real roots; a single, repeated real root; or complex conjugate roots. | https://openstax.org/books/calculus-volume-3/pages/7-key-concepts |
Initial conditions or boundary conditions can then be used to find the specific solution to a differential equation that satisfies those conditions, except when there is no solution or infinitely many solutions. | https://openstax.org/books/calculus-volume-3/pages/7-key-concepts |
To solve a nonhomogeneous linear second-order differential equation, first find the general solution to the complementary equation, then find a particular solution to the nonhomogeneous equation. | https://openstax.org/books/calculus-volume-3/pages/7-key-concepts |
Letyp(x)yp(x)be any particular solution to the nonhomogeneous linear differential equationa2(x)yâ³+a1(x)yâ²+a0(x)y=r(x),a2(x)yâ³+a1(x)yâ²+a0(x)y=r(x),and letc1y1(x)+c2y2(x)c1y1(x)+c2y2(x)denote the general solution to the complementary equation. Then, the general solution to the nonhomogeneous equation is given byy... | https://openstax.org/books/calculus-volume-3/pages/7-key-concepts |
Whenr(x)r(x)is a combination of polynomials, exponential functions, sines, and cosines, use the method of undetermined coefficients to find the particular solution. To use this method, assume a solution in the same form asr(x),r(x),multiplying byxas necessary until the assumed solution is linearly independent of the ge... | https://openstax.org/books/calculus-volume-3/pages/7-key-concepts |
Whenr(x)r(x)isnota combination of polynomials, exponential functions, or sines and cosines, use the method of variation of parameters to find the particular solution. This method involves using Cramerâs rule or another suitable technique to find functionsuâ²(x)uâ²(x)andvâ²(x)vâ²(x)satisfyinguâ²y1+vâ²y2=0uâ²y1â... | https://openstax.org/books/calculus-volume-3/pages/7-key-concepts |
Second-order constant-coefficient differential equations can be used to model spring-mass systems. | https://openstax.org/books/calculus-volume-3/pages/7-key-concepts |
An examination of the forces on a spring-mass system results in a differential equation of the formmxâ³+bxâ²+kx=f(t),mxâ³+bxâ²+kx=f(t),wheremmrepresents the mass,bbis the coefficient of the damping force,kkis the spring constant, andf(t)f(t)represents any net external forces on the system. | https://openstax.org/books/calculus-volume-3/pages/7-key-concepts |
Ifb=0,b=0,there is no damping force acting on the system, and simple harmonic motion results. Ifbâ0,bâ0,the behavior of the system depends on whetherb2â4mk>0,b2â4mk>0,b2â4mk=0,b2â4mk=0,orb2â4mk<0.b2â4mk<0. | https://openstax.org/books/calculus-volume-3/pages/7-key-concepts |
Ifb2â4mk>0,b2â4mk>0,the system is overdamped and does not exhibit oscillatory behavior. | https://openstax.org/books/calculus-volume-3/pages/7-key-concepts |
Ifb2â4mk=0,b2â4mk=0,the system is critically damped. It does not exhibit oscillatory behavior, but any slight reduction in the damping would result in oscillatory behavior. | https://openstax.org/books/calculus-volume-3/pages/7-key-concepts |
Ifb2â4mk<0,b2â4mk<0,the system is underdamped. It exhibits oscillatory behavior, but the amplitude of the oscillations decreases over time. | https://openstax.org/books/calculus-volume-3/pages/7-key-concepts |
Iff(t)â0,f(t)â0,the solution to the differential equation is the sum of a transient solution and a steady-state solution. The steady-state solution governs the long-term behavior of the system. | https://openstax.org/books/calculus-volume-3/pages/7-key-concepts |
The charge on the capacitor in anRLCseries circuit can also be modeled with a second-order constant-coefficient differential equation of the formLd2qdt2+Rdqdt+1Cq=E(t),Ld2qdt2+Rdqdt+1Cq=E(t),whereLis the inductance,Ris the resistance,Cis the capacitance, andE(t)E(t)is the voltage source. | https://openstax.org/books/calculus-volume-3/pages/7-key-concepts |
Power series representations of functions can sometimes be used to find solutions to differential equations. | https://openstax.org/books/calculus-volume-3/pages/7-key-concepts |
Differentiate the power series term by term and substitute into the differential equation to find relationships between the power series coefficients. | https://openstax.org/books/calculus-volume-3/pages/7-key-concepts |
a 2 ( x ) y â³ + a 1 ( x ) y â² + a 0 ( x ) y = r ( x ) a 2 ( x ) y â³ + a 1 ( x ) y â² + a 0 ( x ) y = r ( x ) | https://openstax.org/books/calculus-volume-3/pages/7-key-equations |
a y â³ + b y â² + c y = 0 a y â³ + b y â² + c y = 0 | https://openstax.org/books/calculus-volume-3/pages/7-key-equations |
a 2 ( x ) y â³ + a 1 ( x ) y â² + a 0 ( x ) y = 0 a 2 ( x ) y â³ + a 1 ( x ) y â² + a 0 ( x ) y = 0 | https://openstax.org/books/calculus-volume-3/pages/7-key-equations |
y ( x ) = c 1 y 1 ( x ) + c 2 y 2 ( x ) + y p ( x ) y ( x ) = c 1 y 1 ( x ) + c 2 y 2 ( x ) + y p ( x ) | https://openstax.org/books/calculus-volume-3/pages/7-key-equations |
x â³ + Ï 2 x = 0 x â³ + Ï 2 x = 0 | https://openstax.org/books/calculus-volume-3/pages/7-key-equations |
x ( t ) = c 1 cos ( Ï t ) + c 2 sin ( Ï t ) x ( t ) = c 1 cos ( Ï t ) + c 2 sin ( Ï t ) | https://openstax.org/books/calculus-volume-3/pages/7-key-equations |
x ( t ) = A sin ( Ï t + Ï ) x ( t ) = A sin ( Ï t + Ï ) | https://openstax.org/books/calculus-volume-3/pages/7-key-equations |
m x â³ + b x â² + k x = f ( t ) m x â³ + b x â² + k x = f ( t ) | https://openstax.org/books/calculus-volume-3/pages/7-key-equations |
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