00:01
The point being the spherical coordinates is given as 2 comma 3 pi 5 by 4 comma 5 5 basically the point in spherical coordinates will be of the form r theta and x coordinate is given as r sine theta cost 5, y coordinates are sine theta sine 5, z coordinate is r sine theta, z coordinate is r cost theta, z coordinates are cost theta, z coordinates are cost now when you substitute r as 2 and theta is the second coordinate that is 3 5 by 4, 5 is the third coordinate that is 60 degrees 5 by 3.
01:00
So x value will be 2 times of sine theta, sine 3 5 by 4 which is 1 by root 2 into cost 5 which is cost 60 degrees is half.
01:09
So x value is 1 by root 2.
01:12
Y value will be r sine theta that is 2 into 1 by root 2.
01:15
Now it is sine 5, sine 60 degrees which is root 3 by 2.
01:19
So it is root of 3 by 2.
01:21
The coordinate will be r into cos theta.
01:24
R is 2, pos theta, cos of 3 5 by 4 which is negative 1 by root 2.
01:29
So it is negative root 2.
01:31
So the cartesian coordinates of the point b is basically 1 by root 2, comma root 3 by 2, comma minus root 2.
01:43
So these are the cartesian coordinates.
01:48
Now the next thing is we need to convert the same point p to cylindrical coordinates.
01:53
Remember for cylindrical coordinates the 5 and r should be the same.
02:02
So basically the cylindrical coordinates points will be r comma 5 comma z.
02:07
So z values anyway minus root 2 so r comma 5 comma minus root 2.
02:14
But what is r now? remember r is not equal to 2 here.
02:20
How to find r? instead of r, i'll use the letter r1, so that's to avoid computation, r1.
02:26
Remember, x is equal to r1, fai, y is equal to r1 sine 5.
02:34
Squaring on both sides, i'll get r1 as root of x square plus 1 square.
02:39
Now we have x coordinate as 1 by root, y coordinate is root of 3 by 2, so what will be x square plus y squared? so let's calculate.
02:47
So r1 is equal to r1 is equal to root of x square, 1 by root 2 whole square is half, how, and plus root 3 by 2 whole square is 3 by 2.
02:57
So it is root 2.
02:59
And 5 is same as the phi in the spherical coordinates, that is 5.
03:04
So the cylindrical coordinates will be root 2 comma 5 by 3 comma negative root 2 because z values negative.
03:15
So this is the cylindrical.
03:20
So basically a conclusion is the point b which is 2 comma 3, 5 by 4, 5 by 5, this is spherical, this is the spherical point is equivalent to the point b, 1 by root 2, root of 3 by 2, negative root 2, this is cartesian, cartesian, that is equivalent to root 2, 5 by 3 comma, negative root 2, and this is cylindrical.
03:57
So, this is the conversion.
04:01
So we have some set up questions the next is this scalar field given omegov xyz xy xy xy xy if this two i square x x and vector field v is given and another vector field w is also given so what we need to find is del cross v cross w so basically we need to find curl of v cross product of v and w so that is what we need to find so let's find the v cross -db cross -product of w is given by ijk i'm sorry b cross w right yeah cross part correct so i jk v vector v vector is y plus z z plus x x plus y jvd vector is y z x plus y jvd v x x yvd so let's find the vector determinant so it will be i into z plus x into x y minus x plus y into z x minus j to y plus z into x into x y plus y into x minus x plus y into y z is is k into y plus z into z x minus z plus x into y z so let's simplify so it becomes i times if you can see x is common bracket i'll get y z plus x y x minus x z minus y z minus j into here y is common so you get x y that's x z minus x z minus y z and here z is common we get x z minus y z and here z is common we get x y minus zy minus xy now we have some cancellations all right so we have some cancellations here y z and so is you cancel here x z cancel you cancel or here x y y xx y cancel so we have x square i capped into y minus z minus y square j cap into x minus y square j cap into x minus z plus z squared k -cap into x minus y so basically my v cross w the cross -proved two vector phase will be i cap into x square into y minus z plus j cap into y -square into z -mines x plus k -cap into z -mines x plus k -cap into z -square into x minus y now let's find the curl of it.
07:31
So curl of v cross w will be determinant of ijk, though by o x, though by, though y, do by, do z, x squared into y minus z, y square into z minus x, z, z minus x, z square into x minus y.
07:58
So let's evaluate this determinant very much carefully.
08:06
So i into do by though y of z squared into x minus y will be minus z squared minus d by do by thosey off this is minus y square minus j into do by do x of z squared into x minus y z squared minus y z squared minus du by do z of that will be plus x square.
08:40
So scale cap into do by though x of this will be minus y square minus x squared so finally the kernel of b cross w is minus of y square plus z square into i cap i think you can keep minus outside plus x square plus z square into j cap plus x square plus y square square into k -camp.
09:16
So this is the curl of b cross, double.
09:21
The next question is divergence of v.
09:24
W, you can see the second part, three months.
09:28
What is the dot product? it will be y z into y plus z plus z, plus x, into z plus x, plus x, plus y, now let's find the divergence of this divergence is del dot v dot w, is the divergence, sorry, not del dot, del off.
09:57
It's del off.
10:01
That is gradient.
10:03
So they're asking the gradient of this scale up.
10:06
I'll remove.
10:09
It's del off.
10:10
So del means gradient.
10:13
So this is del.
10:16
So what is del of that? it will be a vector field.
10:19
So it's though by though x off.
10:21
I'll call this some phi.
10:24
So though phi by though x i cap.
10:27
It's do phi by though y, j cap.
10:31
2x2x2x2 k cap what is the partial derivative of the 5 with respect to x it should be z squared plus 2 x plus 2x y plus y square plus y square into i cap i can write that as simply as y square plus x square plus 2x into y plus z i cap by symmetry plus x square plus z square plus two y into x plus z uh j cap plus x square plus y square plus z into x plus y k cap so this is the the gradient of the vector field sorry and scalar field v.
11:37
W that's it.
11:41
Next we need to find the divergence of v cross del omega.
11:45
So let's find del omega.
11:47
Del omega is basically the do by though x of this is y plus 3 z i cap just though by though y will be x plus 4 y j cap plus du by though z will be 3x k k2...