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### Course: Multivariable calculus>Unit 2

Lesson 9: Divergence

# Divergence intuition, part 2

In preparation for finding the formula for divergence, we start getting an intuition for what points of positive, negative and zero divergence should look like. Created by Grant Sanderson.

## Want to join the conversation?

• In general, was the choice of defining this concept as "divergence" arbitrary? I mean, could it have been called "convergence" with positive and negative values corresponding to converging and diverging flow, respectively? Or was there a certain advantage for defining the concept as it is defined now?
• While any mathematical idea could have been defined arbitrarily, they all relate in a certain way that allows them to have certain properties. If it were called "convergence," then the formula would be backwards. The formula connects to other previously established ideas in mathematics, so there is an advantage to calling it "divergence" and not "convergence."

As shown in the next few videos, divergence uses the dot product, so if it were called "convergence," we would have to use the negative value of the dot product.
• Is there a unit for divergence?
• Not intrinsically. When you use a vector field to model something physical, its divergence could be assigned units according to the units of the field.
• Can we describe this problem solution by curl or divergence. Please guide me as I am not expert in mathmatics. Find the constants a, b, and c so that
F=(x+2y+az) i + (bx – 3y – z ) j + (4x + cy + 2 z) k
is irrotational and hence find the function ψ such that F = ∇ ψ
(1 vote)
• If anyone else was wondering how to solve this problem, you can check out how it's done below.

"Irrotational" means that the curl of F is 0.
Remember that the curl of F roughly represents how much rotation there is in F, so you can see how "no rotation in F" or F being "irrotational" would mean that its curl = 0.

Since we know what F is, we can find its curl. We know that the x-component of F is x + 2y + az, the y-component is bx - 3y - z, and the z-component is 4x + cy + 2z. We will call these components:

P(x, y, z) = x + 2y + az = x-component of F
Q(x, y, z) = bx - 3y - z = y-component of F
R(x, y, z) = 4x + cy + 2z = z-component of F

The formula for the curl of F is:

curl F = (∂R/∂y - ∂Q/∂z)i + (∂P/∂z - ∂R/∂x)j + (∂P/∂y - ∂Q/∂x)k

We can calculate the various partial derivatives of P, Q, and R by treating a, b, and c as constants:

∂P/∂y = 2, ∂Q/∂x = b, ∂R/∂x = 4
∂P/∂z = a, ∂Q/∂z = -1, ∂R/∂y = c

So, the curl of F should be:
curl F = (c - -1)i + (a - 4)j + (2 - b)k

Remember that the curl of F must be 0. For a vector to be 0, all of its x-, y-, and z-components must be 0 too. So,

c - -1 = 0 means that c is -1.
a - 4 = 0 means that a is 4.
2 - b = 0 means that b is 2.

Therefore, F must be:
F(x, y, z) = (x + 2y + 4z)i + (2x - 3y - z)j + (4x - y + 2z)k.

The next step is to figure out what scalar-valued function ψ have the gradient F. This would mean that:

∂ψ/∂x = x + 2y + 4z = P
∂ψ/∂y = 2x - 3y - z = Q
∂ψ/∂z = 4x - y + 2z = R

Notice that ∂ψ/∂x only differentiates with respect to x while treating all other variables as constants. Integrating ψ dx will also treat other variables as constants. So, we can get hints at what ψ is like from integrating ψ with respect to x, y, and z separately.

∫ P dx = 1/2 x² + 2xy + 4xz + C
∫ Q dy = -3/2 y² + 2xy - yz + C
∫ R dz = z² + 4xz - yz + C

These antiderivatives don't seem to match, but the C helps us here. As an example, the +C from int ψ dx could include numbers like 2 and 5 but also functions like -yz and z² that don't have x in them. So, the +C allows us to merge what ψ looks like:

ψ(x, y, z) = 1/2 x² - 3/2 y² + z² + 2xy + 4xz - yz + C

The value of C here doesn't matter - it disappears when any derivative of ψ is taken. For convenience, we can let C be 0.

So, we first solved curl F = 0. All components equaling 0 allowed us to solve for a, b, and c. Knowing F, we then integrated it by components to see what ψ should be like. We then concluded that:

ψ(x, y, z) = 1/2 x² - 3/2 y² + z² + 2xy + 4xz - yz