In Exercises 1-8, find the divergence of the field. The gravitational field in Figure 16.9 and E

In Exercises 1-8, find the divergence of the field. The...

Transcript

00:01 Alright, so you have this gravitational field that we will call g is described to minus gmm but these are some constants and then times x along the i direction plus y from the j direction plus c on the k direction divided by x squared plus y squared squared plus y squared plus c squared through the three halves so for convenience let's denote x square plus y square plus c squared to the three halves let's denote that click r so all did i want to compute the divergence of these divergence of g what is that so all that's gonna be you can take out the constant minus gmm partial of my i over r cube xi with respect to x plus partial of y j no without the i plus partial of y over r cube in respect to y plus partial of z over r cube with respect to z over r cube with respect to z so it is equal these so uh so we have then that these divergence of q is equal to um minus m g m so for the first term we have a relative of we have a quotient so then we would have derivative of x if that's x with respect x that's 1 so will be r cube minus x times relative of f with respect partial with respect to x of r cube over r cube squared by the equation rule plus the relative of y with respect y is just one so you see the the equation rule so r cube minus y times partial respect to y of r cube over r cube squared and likewise for c will have our cube minus c comes partial with respect to c of our cube over r of that over our cube squared so that we need to compute to these terms so what is what is this and so well here you have an r cube is equal to x squared plus y squared plus c squared c squared to the three halves and so what is the what is the partial of this with respect to x so so, now we have x times, so these to the same power minus 1 times, oh, it only considers, you only have to consider their term because the rest are constant with respect to x, so you is equal to that, and then this is equal to x.

05:15 So that we will have that this term will be equal to 2 x squared times x squared plus y squared plus c squared today well one is equal to two halves so two halves three halves minus two halves that's one half so that the well this is the partial x times the partial of x and the by in a similar way we have this x squared plus y square plus c squared to the three halves by similar reasons we will have that that would be equal to 2 times y square and then this x squared plus y squared plus t squared two and we also have that this whole z times partial with respect to z of x squared plus y square plus c squared plus c squared to the three halves well but i forgot when we're doing this you should get up three halves right so that it will be that times three halves so that we only get three then also for this one so over here you will get also times three halves and over here you will get that this will be going to c times three halves times two c times x squared plus y squared plus c squared 1 1 half so that for this term here this one is equal to 3 x squared times x squared plus y squared plus c squared 3 1 half and similarly this one would be just 3 times y squared and this one will be 3 times c squared x so that putting together this information we'll have that divergence the divergence of g will be equal to minus g times n times the other m times um so you have here an r cube minus minus shall with respect to x of our cube was equal to 3 x squared times x squared plus y squared plus c squared 3 1 half because of what we did here and so it will be that plus for our cube minus so now for y 3, y squared, x square plus y squared plus c squared to the one half, plus r cubed minus 3 times c squared times x squared plus y squared plus c squared plus c squared plus x squared plus 1 half and all that all those terms divided by r cube and then that square so now we can do some simplifications here you can see that we have three r cubes so that's gonna be minus m gmm so three times r cube and then minus, you can factor this, this is common to all of those, the x squared plus y squared, plus c squared, 2 1 half, so that is multiplying x squared but also y squared and c squared...

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