00:01
Consider a central force vector, f of r, which means that it is a spherically symmetric, so f of r can write as some scalar field f times r hat.
00:17
We want to use the cartesian definition of the curl to show that the curl of a central vector field is equal to zero.
00:30
So on cartesian coordinates, the curl of f is defined as the partial derivative of fx, excuse me, fz, with respect to y, minus the partial derivative of f y with respect to z.
01:23
This is in the i direction, plus the partial derivative of f x with respect to z, minus the partial derivative of f x with respect to z, minus the partial derivative of f z with respect to x it's in the j direction plus the partial derivative of f y with respect to x minus the partial derivative of f x with respect to i why excuse me in the k direction so so now we need to insert our spherically symmetric vector field into this equation.
02:32
The first thing to do is we need to rewrite our polar vector r in cartesian coordinates.
02:49
So we have that r -hap.
02:51
The reason being is because we define the ijk coordinates.
02:56
So r -hat is equal to r -syn theta r.
03:03
Let's use a definition is equal to xx hat plus y y hat plus z z hat divided by d square root of x square plus y square plus d square so now our vector field f r will write as f r times x x hat plus f r times y y hat plus f r times y hat plus f r times z z hat and all this will be divided by r or x square plus y square plus d square two square root now let's compute our partial order so the partial derivative of f x with respect to quantit for y and z so differentiating this coordinate with respect to y y.
04:48
Actually i think it will be here to leave it as r.
05:12
So this corresponds to the partial derivative with respect to y of fr over r times x.
05:25
Y yielding the partial derivative with respect to r of f r over r times the partial derivative of r with respect to y.
05:39
And all this times x because x considered to be a constant.
05:54
Now let's compute the partial derivative r with respect to y.
06:04
And we obtain y over d square roots of x square plus y square plus z squared, in other words, y over r, yielding xy over r times the partial derivative of f r with respect to r times r minus fr all this times one over r squared so now we need the partial derivative of f x with respect to z and we're going to obtain a symmetric result in this case, instead of y, we have r here.
07:22
Z.
07:39
Now we need the partial derivative of fy with respect to x.
07:48
So we've just computed these two terms.
07:52
Now we need these two terms.
08:00
And again, we're going to obtain a very similar result.
08:05
So y here to replace this y.
08:14
Now a factor of x.
08:16
So y times x over r -cube times the partial derivative of f r times r minus f -r and the partial derivative of f -r with respect to z now it gives us y times z over r -cube times the partial derivative of fr with respect to r minus f r and then the partial derivative of f z with respect to x and the partial derivative of f z with respect to y and again we're going to obtain similarly results so z times x over rq times whatever is in this bracket here.
09:27
And the partial due of fz is back to y will give us now z times y over rq...