In the last quiz, we computed the hydrostatic force on the walls of two different aquariums using a new kind of integral: Since the outward unit normal changes from point to point on it is a vector field and so is 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.
Tanks For All The Fish is designing a new home for Gilly the Jellyfish. Everyday she inflates into a full, round sphere of radius and sits at the very bottom of her tank.
To make sure she's comfortable, they need to calculate the total hydrostatic force Gilly feels due to the water inside her.
Place a coordinate system at Gilly's center (which is units below the water's surface with pointing vertically up. Then Here, is the pressure she feels on her equator, and points perpendicularly outward.
What is the correct expression for as a vector field on Gilly's surface
After bringing constants outside the integral, we have We can use symmetry to get us to the right answer quickly.
For example, recall that since there is equal area above and below the curve due to the odd symmetry of
Using symmetry alone, what component(s) of have to be 0?
Now we're in a bind since Gilly is a full sphere which cannot be represented as the graph of a single function like our hemispherical aquarium tank.Through symmetry the total hydrostatic force reduces to
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 as a single piece. Any point in can be described in terms of spherical coordinates If we choose we get every point of in terms of the two angles and This is just what we need! The visualization below shows how varying the two spherical angles in traces out every point on just as traces out every point on
To integrate and find the total hydrostatic force using the sphere representation we need to first work out the in From the aquarium quiz, we know that in terms of Cartesian coordinates
In terms of and however,
Fill in the blank.
Hint: Switching from Cartesian coordinates and to spherical coordinates and requires the use of a Jacobian determinant:
In terms of the spherical angles, we have Use the substitution to evaluate this integral.
We notice something very interesting about our result: Since the volume of a ball of radius can also be written as the triple integral the total hydrostatic force can be written as a triple integral involving the pressure
What is this triple integral given that
By calculating the total hydrostatic force on Gilly's inside, we're led to a rather striking result: It turns out this formula is true if we replace with a general function and think of as a general region in with an outward-pointing unit normal along the boundary surface
In fact, this is just one possible form of the Divergence Theorem! In the final few problems, we'll use the formula to make contact with the form of the Divergence Theorem we saw in the nutshell quiz.
Suppose we have a 3D vector field We can express as a triple integral using the identity What is the integrand? In other words, fill in the blank:
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 -direction and has constant magnitude:
Hint: You could evaluate the integral directly or you could use the Divergence Theorem: is the inside of the sphere As a bonus, try both ways and see that you get the same result.
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.