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paperweight11
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Figured it out! No need to answer. Thanks!
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]
Equations given above.
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!
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!
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