Vector Calculus

In the last quiz, we computed the hydrostatic force on the walls S S of two different aquariums using a new kind of integral:Ftot=Sp n^dA. \vec{F}_{\text{tot}} = \iint_{S} p \ \hat{n}\, dA. Since the outward unit normal n^ \hat{n} changes from point to point on S,S, it is a vector field and so is p n^. p ~ \hat{n}. The total force is therefore a vector field integral.

This quiz continues our work with hydrostatic pressure. Using the surface integral above, we'll uncover one of the greatest results of vector calculus: the Divergence Theorem.

Introducing the Divergence Theorem

                     

Tanks For All The Fish is designing a new home for Gilly the Jellyfish. Everyday she inflates into a full, round sphere of radius R R and sits at the very bottom of her tank.

To make sure she's comfortable, they need to calculate the total hydrostatic force Ftot \vec{F}_{\text{tot}} Gilly feels due to the water inside her.

Place a coordinate system at Gilly's center (which is h h units below the water's surface)) with zz pointing vertically up. Then Ftot=Sp n^dA=p0S[1zh]n^dA. \vec{F}_{\text{tot}} = \iint_{S} p \ \hat{n}\, dA = p_{0} \iint_{S} \left[ 1- \frac{z}{h} \right]\hat{n}\, dA. Here, p0 p_{0} is the pressure she feels on her equator, and n^ \hat{n} points perpendicularly outward.

What is the correct expression for n^(x,y,z) \hat{n}(x,y,z) as a vector field on Gilly's surface S? S ?

Introducing the Divergence Theorem

                     

After bringing constants outside the integral, we have Ftot=Sp n^dA=p0RS[1zh]x,y,zdA. \vec{F}_{\text{tot}} = \iint_{S} p ~ \hat{n}\, d A = \frac{p_{0}}{R} \iint_{S} \left[ 1- \frac{z}{h} \right] \langle x, y, z \rangle\, d A. We can use symmetry to get us to the right answer quickly.

For example, recall that 11x3dx=0 \int_{-1}^{1} x^3 \, dx = 0 since there is equal area above and below the curve y=x3 y = x^{3} due to the odd symmetry f(x)=f(x) f(-x) = - f(x) of f(x)=x3. f(x) =x^3.

Using symmetry alone, what component(s) of Ftot \vec{F}_{\text{tot}} have to be 0?

Introducing the Divergence Theorem

                     

Select one or more

Through symmetry the total hydrostatic force reduces to Ftot=p0RS[1zh]zdAk^. \vec{F}_{\text{tot}} = \frac{p_{0}}{R} \iint_{S} \left[ 1- \frac{z}{h} \right] z \, d A\, \hat{k}. Now we're in a bind since Gilly is a full sphere S, S, which cannot be represented as the graph of a single function like our hemispherical aquarium tank.

We could split poor Gilly along her equator; since both hemispheres are graphs, we can integrate as before and add the two results.

A better option (as Gilly would agree) represents S S as a single piece. Any point in R3 \mathbb{R}^3 can be described in terms of spherical coordinates x=ρsin(ϕ)cos(θ),y=ρsin(ϕ)sin(θ),z=ρcos(ϕ). x = \rho \sin(\phi) \cos(\theta),\quad y = \rho \sin(\phi) \sin(\theta),\quad z = \rho \cos( \phi). If we choose ρ=R, \rho = R, we get every point of S S in terms of the two angles θ \theta and ϕ. \phi. This is just what we need! The visualization below shows how varying the two spherical angles in x(θ,ϕ)=Rsin(ϕ)cos(θ),Rsin(ϕ)sin(θ),Rcos(ϕ) \vec{x}(\theta,\phi) = \big\langle R \sin(\phi) \cos(\theta), R \sin(\phi) \sin(\theta), R \cos( \phi) \big\rangle traces out every point on S,S, just as p+tv \vec{p} + t \vec{v} traces out every point on L. L.

Introducing the Divergence Theorem

                     

Challenge Problem:

To integrate and find the total hydrostatic force using the sphere representation x(θ,ϕ)=Rcos(θ)sin(ϕ),Rsin(θ)sin(ϕ),Rcos(ϕ),  θ[0,2π),  ϕ[0,π], \vec{x}(\theta,\phi) = \big\langle R \cos(\theta) \sin(\phi), R \sin(\theta) \sin(\phi) , R \cos(\phi) \big\rangle,\ \ \theta \in [0,2\pi),\ \ \phi \in [0,\pi], we need to first work out the dA d A in Ftot=p0RS[1zh]zdAk^. \vec{F}_{\text{tot}} = \frac{p_{0}}{R} \iint_{S} \left[ 1- \frac{z}{h} \right] z \, d A \, \hat{k}. From the aquarium quiz, we know that in terms of Cartesian coordinates dA=dxdy1x2+y2R2. d A = \frac{dx\, dy}{\sqrt{1- \frac{x^2+y^2}{R^2}}}.

In terms of θ \theta and ϕ, \phi, however, dA=_____________dθdϕ. dA = \text{\_\_\_\_\_\_\_\_\_\_\_\_\_}\, d \theta\, d \phi.

Fill in the blank.


Hint: Switching from Cartesian coordinates xx and yy to spherical coordinates θ \theta and ϕ\phi requires the use of a Jacobian determinant: dxdy=det(xθxϕyθyϕ)dθdϕ. dx\, dy = \left | \det \begin{pmatrix} \frac{\partial x}{\partial \theta} & \frac{\partial x}{\partial \phi} \\ \frac{\partial y}{\partial \theta} & \frac{\partial y}{\partial \phi} \end{pmatrix} \right |\, d\theta\, d \phi.

Introducing the Divergence Theorem

                     

In terms of the spherical angles, we have Ftot=p0Rθ=0θ=2πϕ=0ϕ=π[1Rcos(ϕ)zh]Rcos(ϕ)z R2sin(ϕ)dϕdθdAk^. \vec{F}_{\text{tot}} = \frac{p_{_0}}{R} \int\limits_{\theta=0}^{\theta = 2\pi}\int\limits_{\phi=0}^{\phi = \pi} \Bigg [ 1- \frac{\overbrace{R\cos(\phi)}^{z}}{h} \Bigg ] \overbrace{R\cos(\phi)}^{z} ~ \overbrace{R^2\sin(\phi)\, d \phi\, d \theta}^{dA} \, \hat{k}. Use the substitution u=cos(ϕ) u = \cos(\phi) to evaluate this integral.

Introducing the Divergence Theorem

                     

We notice something very interesting about our result: Ftot=4πR33Gilly’s volume(p0h)constantk^. \vec{F}_{\text{tot}} = \underbrace{\frac{4 \pi R^3}{3}}_{\text{Gilly's volume}} \overbrace{\left( -\frac{p_{_0}}{h} \right)}^{\text{constant}} \hat{k}. Since the volume of a ball B B of radius R R can also be written as the triple integral 4πR33=B1dxdydz, \frac{4 \pi R^3}{3} = \iiint\limits_{B} 1\, dx\, dy\, dz, the total hydrostatic force can be written as a triple integral involving the pressure p.p.

What is this triple integral given that p=p0[1zh]? p = p_{_0} \left[ 1- \frac{z}{h} \right]?

Introducing the Divergence Theorem

                     

By calculating the total hydrostatic force on Gilly's inside, we're led to a rather striking result: Ftot=Bpdx=Spn^dA. \vec{F}_{\text{tot}} = \iiint\limits_{B} \nabla p\, d\vec{x} = \iint\limits_{S} p\, \hat{n}\, dA. It turns out this formula is true if we replace p p with a general function f, f, and think of B B as a general region in R3 \mathbb{R}^3 with an outward-pointing unit normal n^ \hat{n} along the boundary surface B=S. \partial B = S.

In fact, this is just one possible form of the Divergence Theorem! In the final few problems, we'll use the formula Bfdx=Bfn^dA \iiint\limits_{B} \nabla f\, d\vec{x} = \iint\limits_{\partial B} f\, \hat{n}\, dA to make contact with the form of the Divergence Theorem we saw in the nutshell quiz.

Introducing the Divergence Theorem

                     

Challenge Problem:

Suppose we have a 3D vector field V(x). \vec{V}(\vec{x}). We can express BVn^dA \iint\limits_{\partial B} \vec{V} \cdot \hat{n}\, dA as a triple integral using the identity Bfdx=Bfn^dA. \iiint\limits_{B} \nabla f\, d\vec{x} = \iint\limits_{\partial B} f\, \hat{n}\, dA. What is the integrand? In other words, fill in the blank: BVn^dA=B________________dx. \iint\limits_{\partial B} \vec{V} \cdot \hat{n}\, dA = \iiint\limits_{B} \text{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}\, d\vec{x}.

Introducing the Divergence Theorem

                     

Gilly rests easier in water with a lot of oxygen dissolved in it, so we set up a jet of oxygen-rich water along the bottom of the tank. The jet current points along the xx-direction and has constant magnitude:

Let's write the jet as J=j0i^, \vec{J} = j_{0} \hat{i}, where j0 j_{0} has units of mass (of oxygen) per unit time per unit area. The total rate at which oxygen enters/leaves through Gilly's surface S S per unit time is SJn^dA. \iint\limits_{S} \vec{J} \cdot \hat{n}\, d A. Here S S is the sphere of radius R R centered at the origin. Compute this integral.


Hint: You could evaluate the integral directly or you could use the Divergence Theorem: SJn^dA=BJdx=B[Jxx+Jyy+Jzz]dx. \iint\limits_{S} \vec{J} \cdot \hat{n}\, d A = \iiint\limits_{B} \nabla \cdot \vec{J}\, d \vec{x} = \iiint\limits_{B} \left[ \frac{\partial J_{x}}{\partial x} + \frac{\partial J_{y}}{\partial y} + \frac{\partial J_{z}}{\partial z} \right] d \vec{x}. B B is the inside of the sphere S. S. As a bonus, try both ways and see that you get the same result.

Introducing the Divergence Theorem

                     

We hope you enjoyed this brief intro to the basic ideas of vector calculus.

Our next series of quizzes, Vector-valued Functions, picks up the story of vectors and motion.

Later quizzes develop the divergence and curl vector derivatives as well as their surface integral and line integral counterparts.

The finale brings everything in the course together to solve important real-world problems.

Introducing the Divergence Theorem

                     
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