Calculus based question in my physics course. No real physics required though.

[paperweight11]

Figured it out! No need to answer. Thanks!

Homework Statement

Hey all, I have a question right at the beginning of my fluid mechanics book but it mostly focuses on calculus at this point. This question in particular is all calculus.

a = xyi + y²j + 2k

∇ = ∂/∂x + ∂/∂y + ∂/∂z

What is ∇ . a [DOT PRODUCT of the two]

Homework Equations

Equations given above.

The Attempt at a Solution

I'm assuming since I'm dotting it that the ∂/∂x would be multiplied by the xyi component and so forth for the rest which would mean just do the derivative of each. For some reason the cartesian i j k get dropped in the solutions when they are dotted up. I don't really understand that. When I do the derivative of each I get:

yi + xi + 2yk

The first yi is from doing the derivative of xyi with respect to x, the second xi is from doing the derivative of xyi with respect to y, and the last 2yk is from doing y²j with respect to y.

^hopefully that made sense. The solution manual makes the dotted product drop all the cartesian parts. It goes:


∇.a = ∂ax/∂x + ∂ay/∂y + ∂az/∂z

= y + 2y + 0 = 3y

Not really sure how this works. Would appreciate any help greatly. Thanks!

 

[paperweight11]

Wow never mind I just figured it out by myself a few minutes after posting this! I didn't really understand how these derivatives worked but now I understand that the ∇ = ∂/∂x + ∂/∂y + ∂/∂z also has cartesian i, j, and k and you just match that up with the original and derive accordingly. Got the same answer of 3y. Huh, not too bad! Thanks anyway.