The Jacobian Determinant
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- Опубликовано: 9 янв 2017
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How to interpret the determinant of a Jacobian matrix, along with some examples. 
Хобби
Everytime I begin to hear 3B1B's voice explaining yet another concept I don't know, I know my search is complete
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I found this very helpful in understanding why the Jacobian determinant is needed in expressing multi-integrals via a change of variable. In case this is helpful for others:
1) Change of variables is a transformation of some space. Eg. from R2 to some other R2
2) We can express how the change of variables transformed space via the jacobian matrix. See previous Khan Academy video on the jacobian matrix. Key idea there being the local (small region) effect of any transformation (linear or non-linear) can be expressed by considering the derivative of the transformation function along each input dimension (the basis vectors of the original domain). When the jacobian matrix is multiplied by each basis vector, the resulting vector valued function expresses how each basis vector is transformed. Plugging in a given point then expresses how space is transformed near that point for each original basis vector.
3) The determinant of a matrix is a scalar representing the magnitude of the change in area as a result of the matrix transformation.
4) Therefore the Jacobian determinant is needed in change of variable multi-integrals as it represents what amount the area changed as a result of the variable change
The only thing to watch out is that the transformation is one-to-one ( not one-to-two, for example) and det is non zero for change of variables from (x,y) to (u,v) else the jacobian determinant will not exist of the inverse transformation.
Dear 3blue1brown,
If ever you gonna make a book, I will definitely buy all of it. Please make a book, it will be such a help and will revolutionize the teaching of vector analysis, matrices, etc. Please, thank you.
rather than a book i want him to do a 24hr lecture!
I bet 3blue1brown won't write a textbook, because he succeeds by breaking the bounds of books with computer graphics.
If you would like, you could potentially screenshot portions of the video to print out which you can access like a book
I had to put on Vincent Rubinetti's music in the background to make this video feels right
Hahaha!
haha,i can feel it!!
I don't have words to thank you how much did you helped to me with all these scary concepts that I never understood in the University. Honestly I just want to say: THANK YOU
Thank you! This explains it very well, I was able to grasp it since the very first frame of the video.
Watching this class from BD , proud of you sir.
Thank you so much this is really helpful
Cool but galling to now understand the determinant 10 years after finishing my degree.
thank you so much. Such a beauty it is and the way your taught it
The area of the parallelogram is the cross product of its two sides of vectors( since they all equal to absin(theta), which when calculated is the determinant, that is also how we get the Jacobian from R3 to R3
Thank you very much for this explanation.
How do you cram as much awesomeness as possible into 8 minutes and 52 seconds? THIS.
The video he refers to is: "Local linearity for a multivariable function"
very good and motivating explanation of jacobian
thanks for sharing!
We are so lucky to have 3Brown1Blue as our math teacher, best time to be alive
That is so awesome!!!
Does the Jacobian determinant help you solve PDE's?
This is so beautiful
Thank you brother
Please could you do a video on finding the original value on percentages
You have save the life of a Ugandan with this explanation 🇺🇬♥️
Which software did u use to simulate the determinants?
This is awesome. If only you were my professor when I was in college.
thank you so much my friend!
THANK YOU !
Is there any relation between the Jacobian and taking the normal vector’s magnitude of a parametrised surface????
Is this effectively what makes transformations linear? It seems that det[J[*F*]]=0 means the basis vectors in the output space are redundant and the information about said points is destroyed in the transformation i.e. it is a non-reversible process, which Noether's theorem would probably disallow. Am I right for this thinking?
I can't help but wonder what my college calculus classes were doing freshman year so that I didn't actually know what a matrix was but still used them for calculations
Thank you
better than every other alternative present out there.
Very usefull, thank you
Could you say that the Jacobian matrix is basically this? [del(f1)_transformed del(f2)_transformed]?
This is great, but for infinitesimally small regions. What if I want to find how much a general blob will be squished or stretched by this non-linear transformation if it’s regular sizes not infinitesimally small? Should I do some kind of integrals?
Pretty sure you take the integral of a 2d area evaluated at a to b and inside the integral multiply by the jacobian determinant
how do you calculate Jacobian determinant with 3x3 matrix
Which program are you using to visualise these transformations ?
Kerlyos It’s Python but he has created his own library for it
Where can I find the subject of this video on Khan Academy?
On the Khan Academy website use the search box in the top left and search for "multivarable calculus"
3Brown1Blue
Mrjarnould 3Black1Beige
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What is 3Brown1Blue?
Grant's (the person talking) channel
I knew that voice sounded familiar
Beautiful
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We're truly blessed to have 3b1b😊
Wait that’s the same matrix (at the start) he used in his linear algebra vid
Why in the last example (0,1) when you zoom in it doesn't look like a linear transformation??
It does look like a linear transformation: evenly spaced out parallel lines remain evenly spaced out parallel line
still don't understand how can we use this to do substitution for multiple integral
Why do we need the jacobian for change of variables in an integral
So, I'm not entirely sure if there is or isn't another video covering this, but when you go from, say cartesian coordinates to polar coordinates in, say an integral, then you need to multiply with their jacobian determinants. this is usually found in books like Rottmanns formulae collection, but when finding these, does one then set up a jacobian determinant where f_1 is the cartesian and f_2 is the polar coordinates before the partial differrential?
no i don't think so, since the f_i coordinates represent the functions that take cartesian inputs to polar outputs. one should plug in the polar parameters as f_1 - f_n and then take their partial derivates with respect to the cartesian parameters.
so there may be three functions f_1=r(x,y,z) ; f_2=[theta](x,y,z) and f_3=[phi](x,y,z), which encode the information about how to transform from cartesian to polar. the jacobian will then contain the partial derivatives of these functions with respect to cartesian coordinates, depending on the row and column.
sorry if i did not use the proper mathematical expressions, i'm not a native speaker x)
thanks!
Love you grant....
Why when i compute cos(1) on the calculater it's 0.999
So, does a matrix with determinant 0 everywhere just collapse all space into nothing?
lolzomgz1337 space would be reduced by one dinension, so in the case of the xy plane, it would be reduced to a one dimensional line
He actually did a video about this in his own channel (3Blue1Brown) in the Linear Algebra series. Check it out it makes everything fall into place.
As Matthew said, it would shrink to a lower dimension, but I want to add that it could definitely collapse into nothing, for example in 2d space, a zero determinant means that the area is zero, and a line “1D” definitely satisfies that condition but so does a point in space a “0D” which means it collapsed into nothing if you will.
I wish I'd thought of that question.
great
What is the difference(s) between matrix determinant and Jacobian matrix determinant? Is it only about factor of scale?
I don't know if you still have this question, but a Jacobian matrix is used to perform coordinate transforms. It's just a special kind.
lolzomgz1337 Hi, bud! What do you mean by "perform coordinate transforms"? Greetings from Brazil!
Hey!
The Jacobian is used to map one coordinate system (Eg, 3D Cartesians) to another (Eg, Spherical Polars). So, if you have a vector defined in the 3D Cartesian coordinate system, the Cartesian-Spherical Polar Jacobian will return that vector in terms of Spherical Polar coordinates.
So, the determinant of the Jacobian still does tell you how much a small volume is changed by - it's just like any other matrix.
lolzomgz1337 I think I got the idea. I had thought in the Jacobian only as transformations of Rn to Rn, but you enlightned the subject, cause it extends to Rn -> Rm. May I have your e-mail, if you don't mind? I'm taking a master degree in economics, but there are moments where I feel stuck when things get a little more abstract. Thank you!
Sure thing; Vinproud@gmail.com
I'm a physical scientist, so I can't speak to how often our interpretations would line up, though.
What happen geometrically when the two functions are dependent? (Determinant equals 0)
I know this is late, but it's like compressing the space into a lower dimension. So the xy-plane would get squished into a line for this example. The area is intuitively 0 for that because the area of anything on a line is 0.
This has been fascinating, but I am still not sure what can I do with the Jacobian, I mean I have an idea, but there may be many uses I haven't thought about
One use I have personally dealt with is in the field of MRI research. Say you want to trace how the brain volume changes over time, furthermore, say you want to generalize the changes the brains of two groups of subjects are going through. You would need to account for the baseline differences (different skull volume, different shapes, different height...) that do not directly impact with your study.
Now, a way you could do that would be to warp the brains into a common space: to do this, you require a Jacobian transformation
This is used for changing variables in double and triple integrals. It's also another justification for the change from cartesian coordinates to polar coordinates. For example, suppose you wanna find out the area of a circle. You can use a transformation which maps your cartesian coordinates to polar, (x,y)=(r*cos(t), r*sin(t)). For this example, we have f1(r, t)=r*cos(t)=x and f2(r,t)=r*sin(t)=y. If you compute the jacobian determinant for this vector valued function, as shown in the video, you will get J=r. And because we have dx dy=J dr dt, we get dx dy=r dr dt. This is another justification for the polar coordinates double integrals. You can also get the formula for spherical coordinates by the same reasoning.
Very interesting. Can the Jacobian be used as a "matrix function" to represent non linear transformation in a more linearish way?
+Fernando Franco Félix Yes, if you look at infinitesimal changes of a nonlinear transformation, it should look linear. And that's why you can use the jacobian determinant at very small changes.
Precisely, break any non-linear transformation used to warp a structure to its linear, infinitesimal components
I have a doubt from years now.
What's the "symbolic" different notation between jacobian matrix, determinant of jacobian matrix ,and absolute value of determinant of jacobian matrix
Even en papers even Wikipedia has wrong notation and makes no difference somebody in the world should notice this my God
I think d(x;y)/d(u;v) is a matrix I mean the jacobian matrix then
|d(x;y)/d(u;v)| should be ONLY the determinant of that matrix but still remains to do its absolute value and a swear u can check Wikipedia change variables it's written dxdy=|d(x;y)/d(u;v)|dudv and what if |d(x;y)/d(u;v)|
So now, let's plot the determinant of the Jacobian at every point.
What happens if this Jacobian Determinant gets negative? Is this even possible?
That is analogous to the orientation of the plane being flipped
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Why 3blue1brown?
Does Khan academy ONLY teach math??
you haven't visited their web site, have you?
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hi 3b1b :]