Vector Calculus

Under Pressure: Divergence Theorem

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

This unit 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.

Under Pressure: Divergence Theorem

Intro to Vector Calculus

                     

Under Pressure: 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 \( R \) and sits at the very bottom of her tank.

We want to make sure she's comfortable in her new home, so we need to calculate the total hydrostatic force Gilly feels due to the water inside her. If we place a coordinate system at Gilly's center \((\)which is \( h \) units below the water's surface\()\) with \(z\) pointing vertically up, we have \[ \vec{F}_{\text{tot}} = \iint_{S} p \ \hat{n}\, dA = p_{0} \iint_{S} \left[ 1- \frac{z}{h} \right]\hat{n}\, dA.\] Here, \( p_{0} \) is the pressure she feels on her equator, and \( \hat{n} \) points perpendicularly outward.

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

Under Pressure: Divergence Theorem

Intro to Vector Calculus

                     

Under Pressure: Divergence Theorem

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

Taking a step back to single-variable calculus to see how symmetry can help, recall that \( \int_{-1}^{1} x^3 \, dx = 0 \) has to be true since there is equal area above and below the curve \( y = x^{3} \) due to the odd symmetry \( f(-x) = - f(x) \) of \( f(x) =x^3.\)

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

Under Pressure: Divergence Theorem

Intro to Vector Calculus

                     

Select one or more

Under Pressure: Divergence Theorem

Through symmetry the total hydrostatic force reduces to \[ \vec{F}_{\text{tot}} = \frac{p_{0}}{R} \iint_{S} \left[ 1- \frac{z}{h} \right] z \, d A\, \hat{k}.\] Now we're in a bind since Gilly is a full sphere \( S,\) 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 \( S \) as a single piece. Any point in \( \mathbb{R}^3 \) can be described in terms of spherical coordinates \[ x = \rho \sin(\phi) \cos(\theta),\quad y = \rho \sin(\phi) \sin(\theta),\quad z = \rho \cos( \phi). \] If we choose \( \rho = R, \) we get every point of \( S \) in terms of the two angles \( \theta\) and \( \phi.\) This is just what we need! The visualization below shows how varying the two spherical angles in \[ \vec{x}(\theta,\phi) = \big\langle R \sin(\phi) \cos(\theta), R \sin(\phi) \sin(\theta), R \cos( \phi) \big\rangle \] traces out every point on \(S,\) just as \( \vec{p} + t \vec{v} \) traces out every point on \( L.\)

Under Pressure: Divergence Theorem

Intro to Vector Calculus

                     

Under Pressure: Divergence Theorem

Challenge Problem:

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

In terms of \( \theta \) and \( \phi, \) however, \( dA = \text{_____________}\, d \theta\, d \phi.\)

Fill in the blank.


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

Under Pressure: Divergence Theorem

Intro to Vector Calculus

                     

Under Pressure: Divergence Theorem

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

Under Pressure: Divergence Theorem

Intro to Vector Calculus

                     

Under Pressure: Divergence Theorem

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

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

Under Pressure: Divergence Theorem

Intro to Vector Calculus

                     

Under Pressure: Divergence Theorem

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

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

Under Pressure: Divergence Theorem

Intro to Vector Calculus

                     

Under Pressure: Divergence Theorem

Challenge Problem:

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

Under Pressure: Divergence Theorem

Intro to Vector Calculus

                     

Under Pressure: 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 \(x\)-direction and has constant magnitude:

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


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

Under Pressure: Divergence Theorem

Intro to Vector Calculus

                     

Under Pressure: Divergence Theorem

We hope you enjoyed this brief introduction to the basic ideas underlying vector calculus. Our next chapter, Vector-valued Functions, picks up the story of vectors and motion where we left off in this chapter.

Subsequent chapters develop the vector derivatives divergence and curl as well as their surface integral and line integral counterparts. The course finale uses powerful results (like the Divergence Theorem) proven earlier in the course to solve important real-world problems.

Under Pressure: Divergence Theorem

Intro to Vector Calculus

                     
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