Calculus Volume 3

# Key Equations

Calculus Volume 3Key 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
$∭EdivFdV=∬SF·dS∭EdivFdV=∬SF·dS$