This is an excellent demonstration, it helped me a lot, thanks. One thing I find about youtube mathematics is the empowering ability for the audience to pause the video at will. So, don't be afraid to go more quickly, since those who need more time to grasp a concept can easily pause or replay segments as they wish.
Thank you for the clear explanation. I was faltering earlier today to derive the volume equation using other methods, to me this one is much more intuitive.
You explained clearly and well how to derive the equations for the x, y and z axis. Thank you, this has helped me to partly understand the Schrodinger equation, a task which is personal to me. Keep up the good work.
No need to thank me , I have to thank you for your video ! It helped me a lot! You know , you are one of the few people who actually pronounce those letters correctly and that's good. Mathematics and science connecting people :P
Thank you for the good clarification. Pronunciations, as this, are always best to get elucidation from those who actually speak the original language -- in this case, Greek. I do appreciate your expert input. At last, as cited in the well known expression, some say "po-tay-toe" and others "po-tah-toe" .. actually, come to think about it, I've really never heard anyone say it in the latter way, but, ah well, that is a famous saying that is heard repeated again and again. Thanks much :))
As a greek I can assure you that the pronunciation of φ is (phi) as we say bee an the letter θ is pronounced (thita) as Thi-knes. He pronounced very well
thanks a lot sir, I am driving Collision of gas molecules with wall, and needed the concept of elementary volume of solid angle. you greatly describe this.
It should be noted that, for notation purposes, r is used for the radial length in cylindrical coordinates and p (or roe) is used for the radial length in spherical coordinates.
Can anyone please explain to me why the "Sweeping" part is not just r*d(theta), it looks just like the r*d(phi) case of arc length so why is it different?
Thank you so much. I have a Calculus 3 final in a few hours and needed a review of how to do one of these. You did a great job. One question, do you do differential equations also? Or linear algebra? Wish I had found your videos sooner.
Great video, but I have one question for you though. Why does the variable Phi go from 0 to PI and Theta go from 0 to 2PI? I mean, why can't it be that Phi goes from 0 to 2PI and Theta goes from 0 to PI instead?
One of the sides of the "cube" (the differencial volume) is r*sin(phi). Phi is restricted between 0 and π not only because is geometrically intuitive ("you are creating half a circle and then sweeping it 360º") but also because the sine remains positive. If it was the opposite(phi from 0 to 2Pi and theta from 0 to pi), although it would be intuitive, the sine would enter the negative zone, and the final volume would be zero. In short, is because the differential volume was defined that way. You could still do this way, but the problem would have to be separated in two, to negate the negative sine when phi is between pi and 2pi.
Even with the mistake with angle names, this was the best explanation I've ever seen for spherical coordinates. Thank you very much Chistofboy1 for the great work!
Tomi Fodor I know this is late reply but remember that the arc length is the radial distance times the angle. So in this case, the radius is rsin(phi) and the angle is dtheta.
in modern greek yes you're correct, its pronounced "phee" but the pronunciation usually used in maths "phie" in based on the classical greek pronunciation. as is theta beta pi etc
Imagine drawing a circle (0 to 2*pi). How much do you have to rotate that circle to end up with a sphere? If you rotate it by pi the back end of the circle does half the 'work' as well as the front end, so you end up with a sphere
In other words, think of a circle in the xy-plane. You only need to rotate by pi radians (either through the x- or y-axis - it really doesn't matter due to the circle's symmetry) to get a complete sphere.
You have no idea how cool it is that you put Romans 1:16 in your video. God totally used you to speak to me through MATH! So cool!!! Thanks for the awesome video!
The only limit to change in calculating the volume of a hemisphere would be for the upper-one on the parameter phi -- in this case, phi would range from 0 to pi/2 (see that only the upper-limit changed to half of what it was in the video for the whole sphere). The middle integral then results in -cos(phi), like before, which, when evaluated between these limits, gives us -[0 - 1] or simply 1. The limits of the parameters r and theta stay the same as for the whole sphere. Of course, if you consider that a hemisphere occupies half the volume of the whole sphere, then you can also simply take half of the final result that we derived in the video and get the same answer right away (i.e. 2*pi*R^3/3) :)
+Christofboy1 or you can keep all the limits the same and change the limit of theta from 0 to pi instead of 2pi and that would give you a vertical hemisphere
When you put a sphere of diameter d inside a cube the side of which is d, then the volume of the sphere will be found to be d cubed divided by 1.90986. The area of the sphere is determined by the area of the cube so that 6 times d squared divided by the same factor 1.90986, yields the sphere's area. But, not to worry, the Bronze Age formulations will never lose their appeal! Carry on regardless! The stumbling block of course is that troublesome 6/1.90986. That damned thing turns out to be pi! Ah, pi, pi, pi! You gotta luv it! And not to mention that when d = 6 you get the dreaded 666 which spells 'hex' in Greek arithmetic! But, consider this. the ratio between the cube and the sphere is just as everlasting as the ratio between the circumference of a circle and its diameter. With the volume comparison, the ratio of 1.90986 is in fact superior to pi. With the area comparison, a cube will always have 6 sides, so the special case of pi exists regardless of the value of d.
Is there a mathematical/scientific source that you could cite to formalize the pronunciation of the Greek letters by way of some standardized convention? Usually, I try and stick to the original language pronunciations -- as verified by makisJacob above -- since this is the most authentic way to go, at last.
Aeryl Vales What you described would give you the volume for a semi sphere. Remember: phi is the angle at which you rotate your differential sector about it’s center.
Strange pronunciation / accent. Are you from New Zeland? It certainly is not English! You are using the wrong names for the angles. What you term "Phi" is usually (in most sciences like Physics, Astronomy etc.) denoted by "Theta" and termed "Polar angle". Your angle "theta" is called "Azimuthal angle" or simply "Azimuth". Apart from that, your presentation is good although very long-winded in places. I supposed you assume a low level of mathematical expertise for the majority of the readers ... to be on the safe side. Certainly the exposition is clear and with good graphics.
This is an excellent demonstration, it helped me a lot, thanks.
One thing I find about youtube mathematics is the empowering ability for the audience to pause the video at will. So, don't be afraid to go more quickly, since those who need more time to grasp a concept can easily pause or replay segments as they wish.
Your appreciation is all the reward for me for having prepared the lesson to begin with. Thank you!! :))
I am a tutor , I found this helpful to understand the spherical coordinates and to find the volume of Sphere.
Clear Explanation.... Thank You Very much....I was confused with the “sin” but your element ( larger representation) cleared my confusions...
Thank you for the clear explanation. I was faltering earlier today to derive the volume equation using other methods, to me this one is much more intuitive.
The very best explanation I have come across! Well done and thank you!
Your kind words are all the reward .. thank you very much, and best wishes to you :))
You explained clearly and well how to derive the equations for the x, y and z axis. Thank you, this has helped me to partly understand the Schrodinger equation, a task which is personal to me. Keep up the good work.
No need to thank me , I have to thank you for your video ! It helped me a lot! You know , you are one of the few people who actually pronounce those letters correctly and that's good. Mathematics and science connecting people :P
Thank u
And u taught the differential volume in a very nice way
Truly a great explanation of spherical co-ordinates..
Hey..integrate my thanks from 0 to infinity. Thanks a lot!
Why it is not possible to write
0≤phi≤2π and
0≤theta≤2π
Glad I came by and watched this video. I see now that when I was drawing my volume element, I drew my azimuthal angle all wrong.
Thank you for the good clarification. Pronunciations, as this, are always best to get elucidation from those who actually speak the original language -- in this case, Greek. I do appreciate your expert input. At last, as cited in the well known expression, some say "po-tay-toe" and others "po-tah-toe" .. actually, come to think about it, I've really never heard anyone say it in the latter way, but, ah well, that is a famous saying that is heard repeated again and again. Thanks much :))
As a greek I can assure you that the pronunciation of φ is (phi) as we say bee an the letter θ is pronounced (thita) as Thi-knes. He pronounced very well
thanks a lot sir, I am driving Collision of gas molecules with wall, and needed the concept of elementary volume of solid angle. you greatly describe this.
This is a really good demonstration.
Thanks for the video! The explanation was easy for me to understand.
it was worth watching . that was really really well explained
It should be noted that, for notation purposes, r is used for the radial length in cylindrical coordinates and p (or roe) is used for the radial length in spherical coordinates.
tHIS IS SO INFORMATIVE! i LIKE YOUR DRAWING!
Can anyone please explain to me why the "Sweeping" part is not just r*d(theta), it looks just like the r*d(phi) case of arc length so why is it different?
Because that side is an arc of a circle with radius rsin(theta) and not r
Thank you so much. I have a Calculus 3 final in a few hours and needed a review of how to do one of these. You did a great job. One question, do you do differential equations also? Or linear algebra? Wish I had found your videos sooner.
This was very well done. Thank you.
Nice teacher！
Good ,nice work 👍
This is absolutely perfect! Thanks for thr clear explanation :)
Great video, but I have one question for you though. Why does the variable Phi go from 0 to PI and Theta go from 0 to 2PI? I mean, why can't it be that Phi goes from 0 to 2PI and Theta goes from 0 to PI instead?
One of the sides of the "cube" (the differencial volume) is r*sin(phi). Phi is restricted between 0 and π not only because is geometrically intuitive ("you are creating half a circle and then sweeping it 360º") but also because the sine remains positive. If it was the opposite(phi from 0 to 2Pi and theta from 0 to pi), although it would be intuitive, the sine would enter the negative zone, and the final volume would be zero.
In short, is because the differential volume was defined that way.
You could still do this way, but the problem would have to be separated in two, to negate the negative sine when phi is between pi and 2pi.
@@deox4225 had the same question, thank you!
Even with the mistake with angle names, this was the best explanation I've ever seen for spherical coordinates. Thank you very much Chistofboy1 for the great work!
عاشت ايدك يا ورده 🌹🌹🌹🥰🥰🥰🥰🥰 متعرف شگد ساعدتني
Why fi do not vary 0 to 2pi radian and theta 0 to pi radian?
What happens when you do this on the left side of the sphere? Can this be done with vectors too
Why change in volume = multiplication of all changes?
I don't understand 10:20 - 10:35. Can someone explain why this is true?
Tomi Fodor I know this is late reply but remember that the arc length is the radial distance times the angle. So in this case, the radius is rsin(phi) and the angle is dtheta.
in modern greek yes you're correct, its pronounced "phee" but the pronunciation usually used in maths "phie" in based on the classical greek pronunciation. as is theta beta pi etc
superb explanation!
why phi goes from 0 to Pi and not from 0 to 2*Pi?
Imagine drawing a circle (0 to 2*pi). How much do you have to rotate that circle to end up with a sphere? If you rotate it by pi the back end of the circle does half the 'work' as well as the front end, so you end up with a sphere
In other words, think of a circle in the xy-plane. You only need to rotate by pi radians (either through the x- or y-axis - it really doesn't matter due to the circle's symmetry) to get a complete sphere.
well said!!
why phi is only limited by Pi and not 2Pi?
Is there a mathematical way to prove that phi goes from 0 to pi?
You have no idea how cool it is that you put Romans 1:16 in your video. God totally used you to speak to me through MATH! So cool!!! Thanks for the awesome video!
Thank you!
This is great!
Super helpful!!! Thank you!
very nice explanation..
Thanks for uploading this :) You made it simple to understand :)
Excellent sir,
Can we have how to find volume of solid sphere enclosed by 2 sphere?
so if i was to calculate the vol of a hemisphere, how would phi and theta change?
The only limit to change in calculating the volume of a hemisphere would be for the upper-one on the parameter phi -- in this case, phi would range from 0 to pi/2 (see that only the upper-limit changed to half of what it was in the video for the whole sphere). The middle integral then results in -cos(phi), like before, which, when evaluated between these limits, gives us -[0 - 1] or simply 1. The limits of the parameters r and theta stay the same as for the whole sphere.
Of course, if you consider that a hemisphere occupies half the volume of the whole sphere, then you can also simply take half of the final result that we derived in the video and get the same answer right away (i.e. 2*pi*R^3/3) :)
christofboy1 I would, but I specifically need a hemisphere for some misc calculations involved ;) thanks a lot for your help xD
Illuminati6396 You're most welcome! Best wishes to you.
+Christofboy1 or you can keep all the limits the same and change the limit of theta from 0 to pi instead of 2pi and that would give you a vertical hemisphere
When you put a sphere of diameter d inside a cube the side of which is d, then the volume of the sphere will be found to be d cubed divided by 1.90986. The area of the sphere is determined by the area of the cube so that 6 times d squared divided by the same factor 1.90986, yields the sphere's area. But, not to worry, the Bronze Age formulations will never lose their appeal! Carry on regardless! The stumbling block of course is that troublesome 6/1.90986. That damned thing turns out to be pi! Ah, pi, pi, pi! You gotta luv it! And not to mention that when d = 6 you get the dreaded 666 which spells 'hex' in Greek arithmetic! But, consider this. the ratio between the cube and the sphere is just as everlasting as the ratio between the circumference of a circle and its diameter. With the volume comparison, the ratio of 1.90986 is in fact superior to pi. With the area comparison, a cube will always have 6 sides, so the special case of pi exists regardless of the value of d.
I'm glad the video was helpful to you .. thanks for your nice note :))
🙏 thanks a lot
Very helpful ☺️
Thanks much!! appreciate your time.
Holy shit. You just made my phone screen 3D!!!
Is there a mathematical/scientific source that you could cite to formalize the pronunciation of the Greek letters by way of some standardized convention? Usually, I try and stick to the original language pronunciations -- as verified by makisJacob above -- since this is the most authentic way to go, at last.
Thanks a lot, well explained!
Thanks a lot mate!
Sir isn't phi range between 0 to 90 degrees
Aeryl Vales What you described would give you the volume for a semi sphere. Remember: phi is the angle at which you rotate your differential sector about it’s center.
Ok sir thanks a lot
Why the thita from zreo to pi?
☺️Ok now I know, thankyou so much for your help
Thanks a lot!
Beautiful
thank you sooo much.
My pleasure
Thank youuu very muchh ^^
Thanks !
Nice...
theta should be the angle from the positive z-axis to the point, not phi
Lord my teachers also does not explain in this way...😊😊😊😊😘😍😚🤗
Thanks a lot
thank you
The pleasure is mine.
great thank you
bit too fast for me, but very good
thanks
thankyou!
Thank you!!!
You're totally welcome! Happy studying :))
awsome
ONE QUESTION
Strange pronunciation / accent. Are you from New Zeland? It certainly is not English! You are using the wrong names for the angles. What you term "Phi" is usually (in most sciences like Physics, Astronomy etc.) denoted by "Theta" and termed "Polar angle". Your angle "theta" is called "Azimuthal angle" or simply "Azimuth". Apart from that, your presentation is good although very long-winded in places. I supposed you assume a low level of mathematical expertise for the majority of the readers ... to be on the safe side. Certainly the exposition is clear and with good graphics.
Lol 14:04 - 14:20
Unashamed
thanks