Calculus Volume 3

# Key Equations

Calculus Volume 3Key Equations

### Key Equations

 Vector field in $ℝ2ℝ2$ $F(x,y)=〈P(x,y),Q(x,y)〉F(x,y)=〈P(x,y),Q(x,y)〉$ or $F(x,y)=P(x,y)i+Q(x,y)jF(x,y)=P(x,y)i+Q(x,y)j$ Vector field in $ℝ3ℝ3$ $F(x,y,z)=〈P(x,y,z),Q(x,y,z),R(x,y,z)〉F(x,y,z)=〈P(x,y,z),Q(x,y,z),R(x,y,z)〉$ or $F(x,y,z)=P(x,y,z)i+Q(x,y,z)j+R(x,y,z)kF(x,y,z)=P(x,y,z)i+Q(x,y,z)j+R(x,y,z)k$
 Calculating a scalar line integral $∫Cf(x,y,z)ds=∫abf(r(t))(x′(t))2+(y′(t))2+(z′(t))2dt∫Cf(x,y,z)ds=∫abf(r(t))(x′(t))2+(y′(t))2+(z′(t))2dt$ Calculating a vector line integral $∫CF·ds=∫CF·Tds=∫abF(r(t))·r′(t)dt∫CF·ds=∫CF·Tds=∫abF(r(t))·r′(t)dt$ or $∫CPdx+Qdy+Rdz=∫ab(P(r(t))dxdt+Q(r(t))dydt+R(r(t))dzdt)dt∫CPdx+Qdy+Rdz=∫ab(P(r(t))dxdt+Q(r(t))dydt+R(r(t))dzdt)dt$ Calculating flux $∫CF·n(t)‖n(t)‖ds=∫abF(r(t))·n(t)dt∫CF·n(t)‖n(t)‖ds=∫abF(r(t))·n(t)dt$
 Fundamental Theorem for Line Integrals $∫C∇f·dr=f(r(b))−f(r(a))∫C∇f·dr=f(r(b))−f(r(a))$ Circulation of a conservative field over curve C that encloses a simply connected region $∮C∇f·dr=0∮C∇f·dr=0$
 Green’s theorem, circulation form $∮CPdx+Qdy=∬DQx−PydA,∮CPdx+Qdy=∬DQx−PydA,$ where C is the boundary of D Green’s theorem, flux form $∮CF·dr=∬DQx−PydA,∮CF·dr=∬DQx−PydA,$ where C is the boundary of D Green’s theorem, extended version $∮∂DF·dr=∬DQx−PydA∮∂DF·dr=∬DQx−PydA$
 Curl $∇×F=(Ry−Qz)i+(Pz−Rx)j+(Qx−Py)k∇×F=(Ry−Qz)i+(Pz−Rx)j+(Qx−Py)k$ Divergence $∇·F=Px+Qy+Rz∇·F=Px+Qy+Rz$ Divergence of curl is zero $∇·(∇×F)=0∇·(∇×F)=0$ Curl of a gradient is the zero vector $∇×(∇f)=0∇×(∇f)=0$
 Scalar surface integral $∫∫Sf(x,y,z)dS=∫∫Df(r(u,v))||tu×tv||dA∫∫Sf(x,y,z)dS=∫∫Df(r(u,v))||tu×tv||dA$ Flux integral $∬SF·NdS=∬SF·dS=∬DF(r(u,v))·(tu×tv)dA∬SF·NdS=∬SF·dS=∬DF(r(u,v))·(tu×tv)dA$
 Stokes’ theorem $∫CF·dr=∬ScurlF·dS∫CF·dr=∬ScurlF·dS$
 Divergence theorem $∭EdivFdV=∬SF·dS∭EdivFdV=∬SF·dS$
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