Calculus 3 Lecture 14.8: How to Change Variables in Multiple Integrals (Using the Jacobian)
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- čas přidán 3. 05. 2016
- Calculus 3 Lecture 14.8: How to Change Variables in Multiple Integrals (Using the Jacobian): Just like what it says! What the Jacobian is and how to use it to do substitutions in multiple integrals. Also discussed is how to find your own transformations.
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17:10 Who else said "Jacobian" sitting alone in their room somewhere haha
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54:52 Example on when transformations are NOT given! (I.e. Finding the transformation on your own!)
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Exactly....that's what I'm looking for....did he explain that?.... i.e., (when the transformations aren't given)
Now try to do a problem that is a parallelogram that the transformation is not given.... lol my damn calc teacher man.
Thank you.
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same
because the night before the exam is when we start looking for it :D
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Right at the end, where you calculate the Jacobian for cartesian->spherical transformation, the algebra is a little simpler if you expand the determinant using the bottom row rather than the top row. (One of the 2x2s becomes zero immediately.)
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This is often overlooked but thank you for explaining the answer. I didn't understand what v equaling zero meant in this transformation until you explained.
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I find this totally fascinating, as the Professor made aware of.
In the beginning with our first substitutions, we wanted to change the integrand as the integral was difficult due to them and had to take care to adjust the regions (back in the R1-days just bounds of intervals),
now we want to change the region, as it is now the culprit making the integration difficult, but then we need to take care that we adjust the integrand (which the Jacobian happens to let us do).
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22:38 right after he said "if you are getting wrong answers you forgot the Jacobian" I did a problem, got it wrong, wondered why, and realized I forgot the Jacobian
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Oh, man! That proof for r and p^2sin phi being the Jacobian! So cool
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Towards the end, you converted dxdydz into the spherical co-ordinate equivalent. But, what if I want to convert dxdy, dydz, dzdx into spherical co-ordinate equivalent. How can I convert using Jacobian? Here do we not have 2 variables that depends on 3 other variables?
is the Jacobean related to lambda in Lagrange multipliers or in linear algebra eigenvalues
Professor Leonard ,thank for another beautiful video/ lecture on How to Change Variables in Multiple Integrals ( Using Jacobian) in Calculus Three. This lecture is easy to follow from start to finish, however the transformation and the algebra involved in the change of variables can be problematic and confusing. The examples continue to breakdown each topic from top to bottom and vice versa.
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proof that r is the jacobian for a polar system :
x = r cos theta
y = r sin theta
jacobian = det [ cos theta..........- r sin theta ]
...........................[ sin theta............ r cos theta ]
= cos theta * r cos theta - ( - r sin theta ) ( sin theta )
= r cos^2 theta + r sin^2 theta
= r ( cos^2 theta + sin^2 theta )
= r * 1
= r
Thank you! This part confused me
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Exam is in 5 days, will have to come up with the transformation myself in order to pass. Thank you so much sir.
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what about the jacobian for trig sub ( using the triangle to sub things? )
hey, does anyone know how he got the transformations for the problem starting at ~33:00? I used u=y+2x and v=2y-x (which got me x = (2u-2v)/5 and y = (u+4v)/5) and somehow got the same answer as he did. is it bc the transformation you choose (at least for the parallelograms) really doesn't matter since the Jacobian would make up for any difference?
Hi Professor Leonard! Thank you for your videos, they are really helpful. However, in example (x+y) I never understand how did you get x=u+2v and y=v-2u
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Hello Professor, at [1:22:10] of video while finding the jacobian you took the value of delY over delV as 1 but isn't it supposed to be 2v? please clear me this query as the answer would be wrong if the jacobian is wrong. I'm getting the jacobian determinent as 2
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21:30 why do we not take the absolute value of the Jacobian piece?
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At 1:18:40 how does he know to take the positive square root of u and the positive square root of 2u for his v bounds.