MATH 324 Advanced Multivariable Calculus

Contents
Description Equations
Double integrals Rf(x,y) dA=limm,ni=1mj=1nf(xij,yij)ΔA\iint\limits_{R} f(x, y) \ dA \newline = \lim\limits_{m, n \to\infty} \sum\limits_{i=1}^{m} \sum\limits_{j=1}^{n} f(x_{ij}^{*}, y_{ij}^{*}) \Delta A
Fubini’s Theorem
R=[a,b]×[c,d]R = [a, b]\times[c, d]
Rf(x,y) dA=abcdf(x,y) dx dy=cdabf(x,y) dy dx\iint\limits_{R} f(x, y) \ dA \newline = \int_{a}^{b}\int_{c}^{d} f(x, y) \ dx \ dy \newline = \int_{c}^{d}\int_{a}^{b} f(x, y) \ dy \ dx
Separation of iterative integrals
R=[a,b]×[c,d]R = [a, b]\times[c, d]
Rg(x)h(y) dA=abg(x) dxcdh(y) dy\iint\limits_{R} g(x)h(y) \ dA = \int_{a}^{b} g(x) \ dx\int_{c}^{d} h(y) \ dy
Type I region
D=x×y=[a,b]×[g1(x),g2(x)]D = x \times y = [a, b]\times[g_{1}(x), g_{2}(x)]
Df(x,y) dA=abg1(x)g2(x)f(x,y) dx dy\iint\limits_{D} f(x, y) \ dA = \int_{a}^{b}\int_{g_{1}(x)}^{g_{2}(x)} f(x, y) \ dx \ dy
Type II region
D=x×y=[h1(x),h2(x)]×[c,d]D = x \times y = [h_{1}(x), h_{2}(x)]\times[c, d]
Df(x,y) dA=cdh1(x)h2(x)f(x,y) dy dx\iint\limits_{D} f(x, y) \ dA = \int_{c}^{d}\int_{h_{1}(x)}^{h_{2}(x)} f(x, y) \ dy \ dx
Addition of double integrals D[f(x,y)+g(x,y)] dA=Df(x,y) dA+Dg(x,y) dA\iint\limits_{D} [f(x, y) + g(x, y)] \ dA \newline = \iint\limits_{D} f(x, y) \ dA + \iint\limits_{D} g(x, y) \ dA
Constant multiple of double integrals Dcf(x,y) dA=cDf(x,y) dA\iint\limits_{D} cf(x, y) \ dA = c\iint\limits_{D} f(x, y) \ dA
Region separation of double integrals Df(x,y) dA=D1f(x,y) dA+D2f(x,y) dA\iint\limits_{D} f(x, y) \ dA \newline = \iint\limits_{D_{1}} f(x, y) \ dA + \iint\limits_{D_{2}} f(x, y) \ dA
Area of a region DD A(D)=DdAA(D) = \iint\limits_{D} dA
Average value of a function fˉ=1A(R)Rf(x,y) dA\bar{f} = \dfrac{1}{A(R)} \iint\limits_{R} f(x, y) \ dA
Description Equations
Transformation to polar coordinates x=rcosθy=rsinθx2+y2=r2dA=r dr dθx = r\cos\theta \newline y = r\sin\theta \newline x^{2} + y^{2} = r^{2} \newline dA = r \ dr \ d\theta
Double integrals in polar coordinates
R=r×θ=[a,b]×[α,β]R = r\times\theta = [a, b]\times[\alpha, \beta]
Rf(x,y) dA=αβabf(rcosθ,rsinθ) r dr dθ\iint\limits_{R} f(x, y) \ dA \newline = \int_{\alpha}^{\beta}\int_{a}^{b} f(r\cos\theta, r\sin\theta) \ r \ dr \ d\theta
Double integrals in general polar region
R=r×θ=[h1(θ),h2(θ)]×[α,β]R = r\times\theta = [h_{1}(\theta), h_{2}(\theta)]\times[\alpha, \beta]
Rf(x,y) dA=αβh1(θ)h2(θ)f(rcosθ,rsinθ) r dr dθ\iint\limits_{R} f(x, y) \ dA \newline = \int_{\alpha}^{\beta}\int_{h_{1}(\theta)}^{h_{2}(\theta)} f(r\cos\theta, r\sin\theta) \ r \ dr \ d\theta
Description Equations
Transformation of two variables T(u,v)=(x(u,v),y(u,v))T(u, v) = (x(u, v), y(u, v))
Jacobian of transformation of two variables (x,y)(u,v)=xuxvyuyv\dfrac{\partial (x, y)}{\partial (u, v)} = \begin{vmatrix}\dfrac{\partial x}{\partial u} & \dfrac{\partial x}{\partial v} \cr \cr \dfrac{\partial y}{\partial u} & \dfrac{\partial y}{\partial v}\end{vmatrix}
Change of variables for differentials dA=dxdy=(x,y)(u,v) du dvdA = dx dy = \bigg\lvert \dfrac{\partial (x, y)}{\partial (u, v)} \bigg\rvert \ du \ dv
Change of variables for double integrals Rf(x,y)dA=Sf(x(u,v),y(u,v)) (x,y)(u,v) du dv\iint\limits_{R} f(x, y) dA \newline = \iint\limits_{S} f(x(u, v), y(u, v)) \ \bigg\lvert \dfrac{\partial (x, y)}{\partial (u, v)} \bigg\rvert \ du \ dv
Description Equations
Density function ρ(x,y)=dmdA\rho (x, y) = \dfrac{dm}{dA}
Mass m=Dρ(x,y) dAm = \iint\limits_{D} \rho(x, y) \ dA
Moment about x-axis Mx=Dyρ(x,y) dAM_{x} = \iint\limits_{D} y\rho(x, y) \ dA
Moment about y-axis Mx=Dxρ(x,y) dAM_{x} = \iint\limits_{D} x\rho(x, y) \ dA
Center of mass (xˉ,yˉ)(\bar{x}, \bar{y}) xˉ=Mym=Dxρ(x,y) dADρ(x,y) dA\bar{x} = \dfrac{M_{y}}{m} = \dfrac{\iint\limits_{D} x\rho(x, y) \ dA}{\iint\limits_{D} \rho(x, y) \ dA}

yˉ=Mxm=Dyρ(x,y) dADρ(x,y) dA\bar{y} = \dfrac{M_{x}}{m} = \dfrac{\iint\limits_{D} y\rho(x, y) \ dA}{\iint\limits_{D} \rho(x, y) \ dA}
Moment of inertia about x-axis
(second moment)
Ix=Dy2ρ(x,y) dAI_{x} = \iint\limits_{D} y^{2}\rho(x, y) \ dA
Moment of inertia about y-axis
(second moment)
Iy=Dx2ρ(x,y) dAI_{y} = \iint\limits_{D} x^{2}\rho(x, y) \ dA
Moment of inertia about the origin
(polar moment of inertia)
I0=Ix+Iy=D(x2+y2)ρ(x,y) dAI_{0} = I_{x} + I_{y} = \iint\limits_{D} (x^{2} + y^{2})\rho(x, y) \ dA
Surface area A=D1+(zx)2+(zy)2dAA = \iint\limits_{D} \sqrt{1 + \left( \dfrac{\partial z}{\partial x} \right)^{2} + \left( \dfrac{\partial z}{\partial y} \right)^{2}} dA
Description Equations
Triple integrals Bf(x,y,z) dV=liml,m,ni=1lj=1mk=1nf(xijk,yijk,zijk)ΔV\iiint\limits_{B} f(x, y, z) \ dV \newline = \lim\limits_{l, m, n \to \infty} \sum\limits_{i = 1}^{l} \sum\limits_{j = 1}^{m} \sum\limits_{k = 1}^{n} f(x_{ijk}^{*}, y_{ijk}^{*}, z_{ijk}^{*}) \Delta V
Fubini’s Theorem
$B = x\times y\times z = \newline [a, b]\times[c, d]\times[r, s]$
Bf(x,y,z) dV=rscdabf(x,y,z) dx dy dz=cdrsabf(x,y,z) dx dz dy=\iiint\limits_{B} f(x, y, z) \ dV \newline = \int_{r}^{s}\int_{c}^{d}\int_{a}^{b} f(x, y, z) \ dx \ dy \ dz \newline = \int_{c}^{d}\int_{r}^{s}\int_{a}^{b} f(x, y, z) \ dx \ dz \ dy \newline = …
Type 1 region
D=x×yD = x \times y
E=D×z=D×[u1(x,y),u2(x,y)]E = D \times z = \newline D \times[u_{1}(x, y), u_{2}(x, y)]
Ef(x,y,z) dV=D(u1(x,y)u2(x,y)f(x,y,z)dz)dA\iiint\limits_{E} f(x, y, z) \ dV =\iint\limits_{D} \left( \int_{u_{1}(x, y)}^{u_{2}(x, y)} f(x, y, z) dz \right) dA
Type 2 region
D=y×zD = y \times z
E=D×x=D×[u1(y,z),u2(y,z)]E = D \times x = \newline D \times[u_{1}(y, z), u_{2}(y, z)]
Ef(x,y,z) dV=D(u1(y,z)u2(y,z)f(x,y,z)dx)dA\iiint\limits_{E} f(x, y, z) \ dV =\iint\limits_{D} \left( \int_{u_{1}(y, z)}^{u_{2}(y, z)} f(x, y, z) dx \right) dA
Type 3 region
D=x×zD = x \times z
E=D×y=D×[u1(x,z),u2(x,z)]E = D \times y = \newline D \times[u_{1}(x, z), u_{2}(x, z)]
Ef(x,y,z) dV=D(u1(x,z)u2(x,z)f(x,y,z)dy)dA\iiint\limits_{E} f(x, y, z) \ dV =\iint\limits_{D} \left( \int_{u_{1}(x, z)}^{u_{2}(x, z)} f(x, y, z) dy \right) dA
Example of a general region
(6 general regions)
E=[a,b]×[g1(x),g2(x)]×[u1(x,y),u2(x,y)]E = [a, b]\times[g_{1}(x), g_{2}(x)]\times[u_{1}(x, y), u_{2}(x, y)]
Ef(x,y,z) dV=abg1(x)g2(x)u1(x,y)u2(x,y)f(x,y,z) dz dy dx\iiint\limits_{E} f(x, y, z) \ dV = \int_{a}^{b} \int_{g_{1}(x)}^{g_{2}(x)} \int_{u_{1}(x, y)}^{u_{2}(x, y)} f(x, y, z) \ dz \ dy \ dx
Volume of a solid EE V(E)=EdVV(E) = \iiint\limits_{E} dV
Description Equations
Transformation to cylindrical coordinates x=rcosθy=rsinθz=zx2+y2=r2dV=r dz dr dθx = r\cos\theta \newline y = r\sin\theta \newline z = z \newline x^{2} + y^{2} = r^{2} \newline dV = r \ dz \ dr \ d\theta
Range of cylindrical coordinates r[0,)θ[0,2π]z[0,)r \in [0, \infty) \newline \theta \in [0, 2\pi] \newline z \in [0, \infty)
Triple integrals in general cylindrical region
E=r×θ×z=E = r \times \theta \times z = \newline [α,β]×[h1(θ),h2(θ)]×[u1(x,y),u2(x,y)][\alpha, \beta]\times[h_{1}(\theta), h_{2}(\theta)] \times [u_{1}(x, y), u_{2}(x, y)]
Ef(x,y,z) dV=αβh1(θ)h2(θ)u1(rcosθ,rsinθ)u2(rcosθ,rsinθ)f(rcosθ,rsinθ,z) r dz dr dθ\iiint\limits_{E} f(x, y, z) \ dV = \newline \int_{\alpha}^{\beta} \int_{h_{1}(\theta)}^{h_{2}(\theta)} \int_{u_{1}(r\cos\theta, r\sin\theta)}^{u_{2}(r\cos\theta, r\sin\theta)}… \newline … f(r\cos\theta, r\sin\theta, z) \ r \ dz \ dr \ d\theta
Description Equations
Transformation to spherical coordinates (r=ρ sinϕ)x=ρsinϕcosθy=ρsinϕsinθz=ρcosϕρ2=x2+y2+z2dV=ρ2 sinϕ dρ dθ dϕ(r = \rho \ \sin\phi) \newline x = \rho\sin\phi\cos\theta \newline y = \rho\sin\phi\sin\theta \newline z = \rho\cos\phi \newline \rho^{2} = x^{2} + y^{2} + z^{2} \newline dV = \rho^{2} \ \sin\phi \ d\rho \ d\theta \ d\phi
Range of spherical coordinates ρ[0,)θ[0,2π]ϕ[0,π]\rho \in [0, \infty) \newline \theta \in [0, 2\pi] \newline \phi \in [0, \pi]
Triple integrals in general spherical region
E=θ×ϕ×ρ=E = \theta \times \phi \times \rho = \newline [α,β]×[c,d]×[g1(θ,ϕ),g2(θ,ϕ)][\alpha, \beta] \times [c, d] \times [g_{1}(\theta, \phi), g_{2}(\theta, \phi)]
Ef(x,y,z)dV=cdαβg1(θ,ϕ)g2(θ,ϕ)f(ρsinϕcosθ,ρsinϕsinθ,ρcosϕ)ρ2 sinϕ dρ dθ dϕ\iiint\limits_{E} f(x, y, z) dV = \newline \int_{c}^{d} \int_{\alpha}^{\beta} \int_{g_{1}(\theta, \phi)}^{g_{2}(\theta, \phi)} … \newline … f(\rho\sin\phi\cos\theta, \rho\sin\phi\sin\theta, \rho\cos\phi) … \newline … \rho^{2} \ \sin\phi \ d\rho \ d\theta \ d\phi
Description Equations
Transformation of three variables T(u,v,w)=(x(u,v,w),y(u,v,w),z(u,v,w))T(u, v, w) = (x(u, v, w), y(u, v, w), z(u, v, w))
Jacobian of transformation of three variables (x,y,z)(u,v,w)=xuxvxwyuyvywzuzvzw\dfrac{\partial (x, y, z)}{\partial (u, v, w)} = \begin{vmatrix}\dfrac{\partial x}{\partial u} & \dfrac{\partial x}{\partial v} & \dfrac{\partial x}{\partial w} \cr \cr \dfrac{\partial y}{\partial u} & \dfrac{\partial y}{\partial v} & \dfrac{\partial y}{\partial w} \cr \cr \dfrac{\partial z}{\partial u} & \dfrac{\partial z}{\partial v} & \dfrac{\partial z}{\partial w} \end{vmatrix}
Change of variables for differentials dV=dx dy dz=(x,y,z)(u,v,w) du dv dwdV = dx \ dy \ dz = \bigg\lvert \dfrac{\partial (x, y, z)}{\partial (u, v, w)} \bigg\rvert \ du \ dv \ dw
Change of variables for triple integrals Rf(x,y,z)dV=Sf(x(u,v,w),y(u,v,w),z(u,v,w))(x,y,z)(u,v,w) du dv dw\iiint\limits_{R} f(x, y, z) dV \newline = \iiint\limits_{S} f(x(u, v, w), y(u, v, w), z(u, v, w)) … \newline … \bigg\lvert \dfrac{\partial (x, y, z)}{\partial (u, v, w)} \bigg\rvert \ du \ dv \ dw
Description Equations
Mass m=Eρ(x,y,z) dVm = \iiint\limits_{E} \rho(x, y, z) \ dV
Moments about coordinate planes Myz=Exρ(x,y,z) dVMxz=Eyρ(x,y,z) dVMxy=Ezρ(x,y,z) dVM_{yz} = \iiint\limits_{E} x\rho(x, y, z) \ dV \newline M_{xz} = \iiint\limits_{E} y\rho(x, y, z) \ dV \newline M_{xy} = \iiint\limits_{E} z\rho(x, y, z) \ dV
Center of mass (xˉ,yˉ,zˉ)(\bar{x}, \bar{y}, \bar{z}) xˉ=Myzm=Exρ(x,y,z) dVEρ(x,y,z) dV\bar{x} = \dfrac{M_{yz}}{m} = \dfrac{\iiint\limits_{E} x\rho(x, y, z) \ dV}{\iiint\limits_{E} \rho(x, y, z) \ dV}

yˉ=Mxzm=Eyρ(x,y,z) dVEρ(x,y,z) dV\bar{y} = \dfrac{M_{xz}}{m} = \dfrac{\iiint\limits_{E} y\rho(x, y, z) \ dV}{\iiint\limits_{E} \rho(x, y, z) \ dV}

zˉ=Mxym=Ezρ(x,y,z) dVEρ(x,y,z) dV\bar{z} = \dfrac{M_{xy}}{m} = \dfrac{\iiint\limits_{E} z\rho(x, y, z) \ dV}{\iiint\limits_{E} \rho(x, y, z) \ dV}
Moments of inertia about coordinate axes Ix=E(y2+z2)ρ(x,y,z) dVIy=E(x2+z2)ρ(x,y,z) dVIz=E(x2+y2)ρ(x,y,z) dVI_{x} = \iiint\limits_{E} (y^{2} + z^{2}) \rho(x, y, z) \ dV \newline I_{y} = \iiint\limits_{E} (x^{2} + z^{2}) \rho(x, y, z) \ dV \newline I_{z} = \iiint\limits_{E} (x^{2} + y^{2}) \rho(x, y, z) \ dV
Description Equations
Chain rule
z=f(x(t),y(t))z = f(x(t), y(t))
dzdt=zxdxdt+zydydt\dfrac{dz}{dt} = \dfrac{\partial z}{\partial x}\dfrac{dx}{dt} + \dfrac{\partial z}{\partial y}\dfrac{dy}{dt}
Chain rule
z=f(x(s,t),y(s,t))z = f(x(s, t), y(s, t))
zs=zxxs+zyys\dfrac{\partial z}{\partial s} = \dfrac{\partial z}{\partial x} \dfrac{\partial x}{\partial s} + \dfrac{\partial z}{\partial y} \dfrac{\partial y}{\partial s}

zt=zxxt+zyyt\dfrac{\partial z}{\partial t} = \dfrac{\partial z}{\partial x} \dfrac{\partial x}{\partial t} + \dfrac{\partial z}{\partial y} \dfrac{\partial y}{\partial t}
Chain rule (general)
z=f(x1,,xn)z = f(x_{1}, …, x_{n}),
where xi=xi(t1,,tm)x_{i} = x_{i}(t_{1}, …, t_{m})
zti=zx1xiti++zxnxnti\dfrac{\partial z}{\partial t_{i}} = \dfrac{\partial z}{\partial x_{1}} \dfrac{\partial x_{i}}{\partial t_{i}} + … + \dfrac{\partial z}{\partial x_{n}} \dfrac{\partial x_{n}}{\partial t_{i}}
Description Equations
Assumptions Unit vector u=u1,,ui\mathbf{u} = \langle u_{1}, …, u_{i} \rangle
Independent variables x=x1,,xi\mathbf{x} = \langle x_{1}, …, x_{i} \rangle
General directional derivatives Duf(x)=limh0f(x+hu)f(x)hD_{\mathbf{u}}f(\mathbf{x}) = \lim\limits_{h \to 0} \dfrac{f(\mathbf{x}+h\mathbf{u}) - f(\mathbf{x})}{h}
General gradient vectors f(x)=fx1,,fxi\nabla f(\mathbf{x}) = \bigg\langle \dfrac{\partial f}{\partial x_{1}}, …, \dfrac{\partial f}{\partial x_{i}} \bigg\rangle
General directional derivatives and gradient vectors Duf(x)=f(x)uD_{\mathbf{u}}f(\mathbf{x}) = \nabla f(\mathbf{x}) \cdot \mathbf{u}
Directional derivative in 2D Duf(x,y)=fxcosθ+fysinθD_{\mathbf{u}}f(x, y) = f_{x} \cos\theta + f_{y} \sin\theta
Gradient vector and maximum values max(Duf(x))=f(x)\max(D_{\mathbf{u}}f(\mathbf{x})) = \lvert \nabla f(\mathbf{x}) \rvert
where u=f(x)f(x)\mathbf{u} = \dfrac{\nabla f(\mathbf{x})}{\lvert \nabla f(\mathbf{x}) \rvert}
Gradient vector \perp tangent vector
for level surface F(x,y,z)=kF(x, y, z) = k
F(x(t),y(t),z(t))r(t)=0F(x0,y0,z0)r(t0)=0\nabla F(x(t), y(t), z(t)) \cdot \mathbf{r}'(t) = 0 \newline \nabla F(x_{0}, y_{0}, z_{0}) \cdot \mathbf{r}'(t_{0}) = 0
Tangent plane in terms of gradient vector (normal vector) F(x,y,z)xx0,yy0,zz0=0\nabla F(x, y, z) \cdot \langle x-x_{0}, y-y_{0}, z-z_{0} \rangle= 0
Symmetric equation of normal line xx0Fx(x0,y0,z0)=yy0Fy(x0,y0,z0)=zz0Fz(x0,y0,z0)\dfrac{x-x_{0}}{F_{x}(x_{0}, y_{0}, z_{0})} = \newline \dfrac{y-y_{0}}{F_{y}(x_{0}, y_{0}, z_{0})} = \newline \dfrac{z-z_{0}}{F_{z}(x_{0}, y_{0}, z_{0})}
Description Equations
Vector field F(x)=P(x),Q(x),R(x)\mathbf{F}(\mathbf{x}) = \langle P(\mathbf{x}), Q(\mathbf{x}), R(\mathbf{x}) \rangle
Conservative vector field
and potential function
F=f\mathbf{F} = \nabla f
Parameterization of line segments r(t)=(1t)r0+t r1\mathbf{r}(t) = (1-t) \mathbf{r}_{0} + t \ \mathbf{r}_1, for 0t10 \le t \le 1
Parameterization of circles r(t)=rcos(t),rsin(t)\mathbf{r}(t) = \langle r\cos(t), r\sin(t) \rangle
Parameterization of functions r(t)=t,f(t)\mathbf{r}(t) = \langle t, f(t) \rangle
Arc length L=ab r(t) dtL = \int_{a}^{b} \ \lvert \mathbf{r}'(t) \rvert \ dt
Arc length parameter s(t)=at r(u) dus(t)=r(t)ds=r(t) dts(t) = \int_{a}^{t} \ \lvert \mathbf{r}'(u) \rvert \ du \newline s'(t) = \lvert \mathbf{r}'(t) \rvert \newline ds = \lvert \mathbf{r}'(t) \rvert \ dt
Description Equations
Line integral Cf(x,y) ds=limni=1nf(xi,yi)Δsi\int_{C} f(x, y) \ ds = \lim\limits_{n \to \infty} \sum\limits_{i=1}^{n} f(x_{i}^{*}, y_{i}^{*}) \Delta s_{i}
Line integral with respect to arc length Cf(x,y) ds=abf(x(t),y(t))(dxdt)2+(dydt)2dt=abf(r(t)) r(t) dt\int_{C} f(x, y) \ ds \newline = \int_{a}^{b} f(x(t), y(t)) \sqrt{\left( \dfrac{dx}{dt} \right)^{2} + \left( \dfrac{dy}{dt} \right)^{2}} dt \newline = \int_{a}^{b} f(\mathbf{r}(t)) \ \lvert \mathbf{r}'(t) \rvert \ dt
Line integral with respect to xx and yy Cf(x,y) dx=abf(x(t),y(t)) x(t) dtCf(x,y) dy=abf(x(t),y(t)) y(t) dt\int_{C} f(x, y) \ dx = \int_{a}^{b} f(x(t), y(t)) \ x'(t) \ dt \newline \int_{C} f(x, y) \ dy = \int_{a}^{b} f(x(t), y(t)) \ y'(t) \ dt
Line integrals of vector fields CFT ds=CFdr=abF(r(t))r(t) dt\int_{C} \mathbf{F} \cdot \mathbf{T} \ ds \newline = \int_{C} \mathbf{F} \cdot d\mathbf{r} \newline = \int_{a}^{b} \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t) \ dt
Line integrals of vector fields and scalar fields F=P,Q,RCFdr=CP dx+Q dy+R dz=CP x(t)+Q y(t)+R z(t) dt\mathbf{F} = \langle P, Q, R \rangle \newline \int_{C} \mathbf{F} \cdot d\mathbf{r} \newline = \int_{C} P \ dx + Q \ dy + R \ dz \newline = \int_{C} P \ x'(t) + Q \ y'(t) + R \ z'(t) \ dt
Orientation properties of line integrals with respect to arc length, xx, and yy Cf(x,y) ds=Cf(x,y) dsCf(x,y) dx=Cf(x,y) dxCf(x,y) dy=Cf(x,y) dy\int_{-C} f(x, y) \ ds = \int_{C} f(x, y) \ ds \newline \int_{-C} f(x, y) \ dx = -\int_{C} f(x, y) \ dx \newline \int_{-C} f(x, y) \ dy = -\int_{C} f(x, y) \ dy
Orientation properties of line integrals of vector fields CFdr=CFdr\int_{-C} \mathbf{F} \cdot d\mathbf{r} = -\int_{C} \mathbf{F} \cdot d\mathbf{r}
Line integral of a piecewise-smooth curve Cf ds=i=1NCif dsC=C1C2CN\int_{C} f \ ds = \sum\limits_{i=1}^{N} \int_{C_{i}} f \ ds \newline C = C_{1} \cup C_{2} \cup … \cup C_{N}
  • path - a smooth curve with initial and terminal point
  • simple curve - a curve that does not intersect itself anywhere between its endpoints
  • closed curve - a curve where its terminal point coincides with its initial point
  • simple region - a region that is bounded by two line segments in one direction (type-1, type-2 regions)
  • open region - a region that does not contain boundary points
  • closed region - a region that contains all boundary points
  • connected region - two points in the region can be joined by a path that lies in the region
  • simply-connected region - a region that every simple closed curve in D encloses only points that are in D
    • has no hole
    • doesn’t consist of separate pieces
Description Equations
Fundamental theorem of line integrals Cfdr=f(r(b))f(r(a))\int_{C} \nabla f \cdot d\mathbf{r} = f(\mathbf{r}(b)) - f(\mathbf{r}(a))
Line integrals of non-conservative fields are not path independent (same end points) C1FdrC2Fdr\int_{C_{1}} \mathbf{F} \cdot d\mathbf{r} \not= \int_{C_{2}} \mathbf{F} \cdot d\mathbf{r}
Line integrals of conservative fields are path independent (same end points) C1fdr=C2fdr\int_{C_{1}} \nabla f \cdot d\mathbf{r} = \int_{C_{2}} \nabla f \cdot d\mathbf{r}
Line integrals of closed path CclosedFdr=0CFdr\int_{C_{\mathrm{closed}}} \mathbf{F} \cdot d\mathbf{r} = 0 \Leftrightarrow \newline \int_{C} \mathbf{F} \cdot d\mathbf{r} is path independent
Path independence and conservative vector field (open, connected region) CFdr\int_{C} \mathbf{F} \cdot d\mathbf{r} is path independent \Rightarrow
F=f\mathbf{F} = \nabla f (conservative field)
Property of conservative vector field F=fPy=Qx\mathbf{F} = \nabla f \Rightarrow \dfrac{\partial P}{\partial y} = \dfrac{\partial Q}{\partial x}
Determine conservative vector field in 2D
(open simply-connected region)
Py=QxF=f\dfrac{\partial P}{\partial y} = \dfrac{\partial Q}{\partial x} \Rightarrow \mathbf{F} = \nabla f
  • Fundamental theorem of line integral (FTL) is always true (with assumptions).
  • Other statements are not true for general F=P,Q\mathbf{F} = \langle P, Q \rangle
    • They have to be verified for each given F\mathbf{F} or derived from theorems. curl F=0(checking 3D conservative field)Py=Qx(checking 2D conservative field)open, simply-connected DF=f(def. of conservative field)Cfdr=f(r(b))f(r(a))(FTL)open, connected DC1Fdr=C2Fdr(path independence)CFdr=0 on a closed path(closed path)\begin{aligned} \mathrm{curl}\ \mathbf{F} &= \mathbf{0} &\color{gray}\footnotesize\text{(checking 3D conservative field)} \cr \dfrac{\partial P}{\partial y} &= \dfrac{\partial Q}{\partial x} &\color{gray}\footnotesize\text{(checking 2D conservative field)} \cr & \color{blue}\upharpoonleft\downharpoonright \footnotesize\text{open, simply-connected }D \cr \mathbf{F} &= \nabla f &\color{gray}\footnotesize\text{(def. of conservative field)} \cr \textstyle\int_{C} \nabla f \cdot d\mathbf{r} & = f(\mathbf{r}(b)) - f(\mathbf{r}(a)) &\color{gray}\footnotesize\text{(FTL)} \cr \color{blue}\footnotesize\text{open, connected }D & \color{blue}\upharpoonleft\downharpoonright \cr \textstyle\int_{C_{1}} \mathbf{F} \cdot d\mathbf{r} &= \textstyle\int_{C_{2}} \mathbf{F} \cdot d\mathbf{r} &\color{gray}\footnotesize\text{(path independence)} \cr & \color{blue}\upharpoonleft\downharpoonright \cr \textstyle\int_{C} \mathbf{F} &\cdot d\mathbf{r} = 0 \footnotesize\text{ on a closed path} &\color{gray}\footnotesize\text{(closed path)} \end{aligned}
Description Equations
Gradient grad f=f=fx,fy,fz\mathrm{grad}\ f = \nabla f \newline = \langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \rangle
Curl curl F=×F=RyQz,PzRx,QxPy\mathrm{curl}\ \mathbf{F} = \nabla \times \mathbf{F} \newline = \langle \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \rangle
Divergence div F=F=Px+Qy+Rz\mathrm{div}\ \mathbf{F} = \nabla \cdot \mathbf{F} \newline = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}
Laplace operator on scalar functions 2f=f=div(f)=2fx2+2fy2+2fz2\nabla^{2}f = \nabla\cdot\nabla f = \mathrm{div}(\nabla f) \newline = \frac{\partial^{2}f}{\partial x^{2}} + \frac{\partial^{2}f}{\partial y^{2}} + \frac{\partial^{2}f}{\partial z^{2}}
Laplace operator on vector fields 2F=2P,2Q,2R\nabla^{2}\mathbf{F} = \langle \nabla^{2}P, \nabla^{2}Q, \nabla^{2}R \rangle
Property of conservative vector field curl f=0\mathrm{curl}\ \nabla f = \mathbf{0}
Determine conservative vector field in 3D
(open simply-connected region)
curl F=0F=f\mathrm{curl}\ \mathbf{F} = \mathbf{0} \Rightarrow \mathbf{F} = \nabla f
Property of divergence and curl div curl F=(×F)=0\mathrm{div}\ \mathrm{curl}\ \mathbf{F} = \nabla\cdot(\nabla\times\mathbf{F}) = 0
Determine curl field div F0F\mathrm{div}\ \mathbf{F} \not= 0 \Rightarrow \mathbf{F} is not curl of any field
Description Equations
Green’s Theorem
(positively oriented, piecewise-smooth, simple, closed curve CC enclosing DD)
CFdr=D(QxPy)dA\displaystyle\oint_{C} \mathbf{F}\cdot d\mathbf{r} = \iint\limits_{D} \left( \dfrac{\partial Q}{\partial x} - \dfrac{\partial P}{\partial y} \right) dA
Circulation-Curl form of Green’s Theorem CFdr=D(×F)k dA\oint_{C} \mathbf{F}\cdot d\mathbf{r} = \iint\limits_{D} (\nabla\times\mathbf{F})\cdot \mathbf{k} \ dA
Flux-Divergence form of Green’s Theorem CFn ds=DF dA\oint_{C} \mathbf{F}\cdot \mathbf{n} \ ds = \iint\limits_{D} \nabla\cdot\mathbf{F} \ dA
Area of region DD enclosed by CC A=D1 dA=Cx dy=Cy dx=12Cx dyy dxA = \iint\limits_{D}1\ dA \newline = \oint_{C} x \ dy \newline = -\oint_{C} y \ dx \newline = \dfrac{1}{2}\oint_{C} x \ dy - y \ dx
Description Equations
General parametric surface r(u,v)=x(u,v),y(u,v),z(u,v)\mathbf{r}(u, v) = \langle x(u, v), y(u, v), z(u, v) \rangle
Parametric equation of a plane r(u,v)=r0+ua+vb\mathbf{r}(u, v) = \mathbf{r}_{0} + u\mathbf{a} + v\mathbf{b}
Parametric equation of a sphere r(ϕ,θ)=asinϕcosθ,asinϕsinθ,acosϕ\mathbf{r}(\phi, \theta) = \langle a\sin\phi\cos\theta, a\sin\phi\sin\theta, a\cos\phi \rangle
Parametric equation of an explicit function r(u,v)=u,v,f(u,v)\mathbf{r}(u, v) = \langle u, v, f(u, v) \rangle
Parametric equation of a surface of revolution r(u,θ)=u,f(u)cosθ,f(u)sinθ\mathbf{r}(u, \theta) = \langle u, f(u)\cos\theta, f(u)\sin\theta \rangle
Normal vector of a tangent plane ru×rv\mathbf{r}_{u} \times \mathbf{r}_v
Surface area of a parametric surface Dru×rv dA\iint\limits_{D} \lvert \mathbf{r}_{u}\times\mathbf{r}_{v} \rvert \ dA
Surface area of the graph of an explicit function z=f(x,y)z = f(x, y) D1+(zx)2+(zy)2 dA\iint\limits_{D} \sqrt{1 + (\frac{\partial z}{\partial x})^{2} + (\frac{\partial z}{\partial y})^{2}} \ dA
Surface area of the graph of an implicit function C=f(x,y,z)C = f(x, y, z) Dffk dA\iint\limits_{D} \dfrac{\lvert \nabla f \rvert}{\lvert \nabla f \cdot \mathbf{k} \rvert} \ dA
Description Equations
Surface integral of a function over a parametric surface Sf(x,y,z) dS=Df(r(u,v))ru×rv dA\iint\limits_{S} f(x,y,z) \ dS \newline = \iint\limits_{D} f(\mathbf{r}(u,v)) \lvert \mathbf{r}_{u}\times\mathbf{r}_{v} \rvert \ dA
Surface integral of an explicit function z=f(x,y)z = f(x, y) Sf(x,y,z) dS=Df(x,y,g(x,y))1+(zx)2+(zy)2 dA\iint\limits_{S} f(x,y,z) \ dS \newline = \iint\limits_{D} f(x, y, g(x, y)) \sqrt{1+(\frac{\partial z}{\partial x})^{2}+(\frac{\partial z}{\partial y})^{2}} \ dA
Surface integral of piecewise smooth surface Sf dS=i=1NSif dSS=S1S2SN\iint\limits_{S} f \ dS = \sum\limits_{i=1}^{N}\iint\limits_{S_{i}}f \ dS \newline S = S_{1} \cup S_{2} \cup … \cup S_{N}
Unit normal vector n=ru×rvru×rv\mathbf{n} = \dfrac{\mathbf{r}_{u}\times\mathbf{r}_{v}}{\lvert \mathbf{r}_{u}\times\mathbf{r}_{v} \rvert}
Surface integral of a vector field over a parametric surface SFdS=SFn dS=DF(ru×rv) dA=D(PgxQgy+R)dA\iint\limits_{S} \mathbf{F}\cdot d\mathbf{S} = \iint\limits_{S} \mathbf{F}\cdot \mathbf{n} \ dS \newline = \iint\limits_{D} \mathbf{F}\cdot (\mathbf{r}_{u}\times\mathbf{r}_{v}) \ dA \newline = \iint\limits_{D} \left( -P \frac{\partial g}{\partial x} -Q \frac{\partial g}{\partial y} + R \right) dA
Description Equations
Stoke’s Theorem
(S: oriented, piecewise-smooth surface
C: simple, closed, piecewise-smooth curve
F: continuous partial derivatives in open region)
CFdr=S(×F)dS\displaystyle\int_{C} \mathbf{F}\cdot d\mathbf{r} = \iint\limits_{S} (\nabla\times\mathbf{F})\cdot d\mathbf{S}
Alternative surface
(CC is a common curve of S1S_{1} and S2S_{2})
S1(×F)dS=CFdr=S2(×F)dS\iint\limits_{S_{1}} (\nabla\times\mathbf{F})\cdot d\mathbf{S} \newline = \int_{C} \mathbf{F}\cdot d\mathbf{r} \newline = \iint\limits_{S_{2}} (\nabla\times\mathbf{F})\cdot d\mathbf{S}
Description Equations
Divergence Theorem
(E: simple, solid region
S: positively oriented surface
F: continuous partial derivatives in open region)
SFdS=EF dV\displaystyle\iint\limits_{S}\mathbf{F}\cdot d\mathbf{S} = \iiint\limits_{E}\nabla\cdot\mathbf{F} \ dV
Function Type Domain \to Range Equation Example
Function of several variables RnR\R^{n} \to \R f(x)f(\mathbf{x}) f(x,y,z)=2x2+ey5z37f(x, y, z) = \newline 2x^{2} + e^{y} - 5z^{3} - 7
Vector-valued function RRn\R \to \R^{n} v(t)\mathbf{v}(t) v(t)=t2,2t,et\mathbf{v}(t) = \newline \langle t^{2}, -2t, e^{t} \rangle
Vector field RnRn\R^{n} \to \R^{n} F(x)\mathbf{F}(\mathbf{x}) F(x,y,z)=3xy,z,z2x\mathbf{F}(x, y, z) = \newline \langle 3x-y, z, z^{2}-x \rangle