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### Course: Multivariable calculus>Unit 2

Lesson 13: Jacobian

# Local linearity for a multivariable function

A visual representation of local linearity for a function with a 2d input and a 2d output, in preparation for learning about the Jacobian matrix.

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• Its similar to the fact that the earth can look flat to humans because we are close to the surface but that's because the earth far larger than us. Many, if not most, continuous and smooth functions / topology would have this property at the limit, correct? as long as the resulting topology after the transformation is smooth.
• I believe you are correct. The closer you zoom in on something the less of that total change you are seeing which is why it appears flat. It's as if you are taking the limit of something and getting closer and closer to a certain number without ever getting there.
• 1. Do all linear transforming functions have jacobians?
2. If functions perform non-linear transformation, are jacobians approximations? If so, can we express it as y(x+dx) - y(x) = Jdx + error? How big is the error in most cases?
• 1) Linear functions (those that verify f(x+y) = f(x)+f(y) and f(c*x) = c*f(x) where c is a scalar and x and y are vectors) are the same, in finite dimension, as your matrices with "only numbers in them" (no variables in the coefficients). All of them have a Jacobian, yes, and they are a very special case, their Jacobian is equal to themselves in every point of the space. This means [[4,3],[5,-6]] for example, is a linear function (the dimension m*n of the matrix represents that it's a linear function from R^n to R^m, or C^n to C^m for matrices with complex numbers as coefficients, so here our 2*2 example is from R² to R²). And the jacobian (the "true" multivariate generalization of our classical derivative) is also the matrix [[4,3],[5,-6]].

For R¹ to R¹ functions, our usual derivative f'(x) can technically be understood as a 1*1 matrix. When you take for example the 1D to 1D linear function f = x -> 4x, which takes the "1D vector" x and returns the 1D vector 4x, it could be understood as the 1*1 matrix [4]. Its derivative in every point is the number 4, which can also be understood as a function rather than a number: the 1*1 matrix [4].

2) Remember series expansion of functions ? Where you use the Taylor formula which spans as many derivatives as it can, and has an integral remainder which is negligible when close to a certain point ? Well it can be generalized for multiple variables as well, http://fourier.eng.hmc.edu/e176/lectures/NM/node45.html
Is this what you meant ? The size of the error will completely depend on your dx and second/third/fourth etc multivariate derivatives, which are tensors (hypermatrices) if I'm not mistaken.
• Do all multi variable transformations of 2-d planes onto other 2-d planes posses local linearity?
• This is my guess, I think there's local linearity at a point (𝑥₀, 𝑦₀) if the function is differentiable at the point. This idea is an extension of that you can linearly approximate a single-variable scalar function at a point (𝑥₀, 𝑦₀) when the function is differentiable at the point.
This is not a rigorous proof, maybe there must be different things you should consider.
• Which software is the one used to make the Khan multivariable videos?
• Grant uses Manim, an animating software that he developed for his videos, to make the animated graphs and vector fields.
• Is locally linear the same as continuous?
(1 vote)
• Not really ... think of it like a tangent approximation.