Teaching Calculus

Site owners

  • Nikolay Brodskiy
  • Alexander Brodsky

Transcript

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So this is what we understood about a function from a plane to a plane.

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Now the challenging question is to understand ...

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... the functions from 3-space to 3-space.

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And physics interpretation of such a function could be the vector field in 3-space.

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When each point in 3-space gains a vector, a velocity vector.

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And then of course the similar type of simple functions ...

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... linear functions could be considered at first.

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And ...

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A linear function would be represented by a matrix which is going to be three by three.

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And the question would be ...

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... is it true that any linear flow of media in 3-space ...

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... can be decomposed into the sum of three relatively simple flows like gas ...

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... and liquid ...

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... and solid.

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And then if yes then how to describe those matrices?

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What kind of matrices identify the gas behavior ...

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... liquid behavior and solid behavior?

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And ...

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Well just a curious question would be ...

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... to see the ...

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... the number of parameters [...] to describe each of these behaviors.

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For example for the gas behavior here ...

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... we understood that it's ...

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... it depends on one parameter only.

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On this rate ...

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... of expansion.

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And algebraically it's just a coefficient with the identity matrix.

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So it is one parameter family.

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The circulation ...

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... is completely described by ...

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... one coefficient with that matrix.

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But liquid behavior is described by two coefficients.

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l one -- the coefficient with ...

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... that matrix plus l two ...

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... times zero, one, one, zero.

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So in some sense ...

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... the gas and solid types are one-dimensional and ...

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... liquid type is two-dimensional.

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So that the sum of all these dimensions makes the dimension of all the matrices -- dimension four.

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Dimension of all those two by two matrices.

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Now in 3 by 3 case we have 9 -- the dimension of all matrices.

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So if we are to observe the decomposition in three types of behaviors ...

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... then we should expect the sum of those dimensions, the sum of corresponding dimensions ...

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... to be nine. And then the question would be well ...

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... how many dimensions, how many parameters are there ...

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... in the gas, liquid, or solid types?

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Now er ...

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Actually once you understand that there is no difficulty ...

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... extending all that two functions ...

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... from ...

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... four-dimensional space to four-dimensional space.

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And actually even further ...

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... functions from n-dimensional Euclidean space to n-dimensional Euclidean space.

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And this kind of functions can be interpreted as ...

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... vector fields in the corresponding space.

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When each point gains a velocity vector assigned.

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Now the question is yes, yes, we can go into those higher dimensions ...

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... carrying out all the algebra, using this ...

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... formalism of matrices.

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Now what is the point? I mean why would anybody want to understand ...

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... the flow of gas, or liquid, or solid in four-dimensional space?

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In three-dimensional space yes.

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We live in three-dimensional space. You want to understand it.

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But in four-dimensional space, in five-dimensional space?

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Now there is a big question -- why?

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Why would we want ...

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... to study all that?

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