Multi-variable systems?

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Re: Multi-variable systems?

Postby joe » Wed Jul 15, 2015 11:26 am

I don't mind calculus being discussed here. Within limits, I don't mind helping out MIT. But, because it is a wealthy, well-resourced institution, I'd rather not make a habit of it.

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Re: Multi-variable systems?

Postby Boris Lagutin » Wed Jul 15, 2015 12:02 pm

Ok. I agree with you :)
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Re: Multi-variable systems?

Postby Boris Lagutin » Wed Jul 29, 2015 7:21 am

Lagrange Multiplier is applied to find maximum and minimum when there is a function and a constraint. However, gradients of the function and the constraint must be parallel at a point in which the function equals the constraint (touching each other). For example, there are a function f(x,y,z) and a constraint g(x,y,z). What if their gradients are not parallel to each other at the point of their touch? May there be such situations?

Thank you.
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Re: Multi-variable systems?

Postby a.s.h. » Fri Jul 31, 2015 8:37 pm

Boris Lagutin wrote:Lagrange Multiplier is applied to find maximum and minimum when there is a function and a constraint. However, gradients of the function and the constraint must be parallel at a point in which the function equals the constraint (touching each other). For example, there are a function f(x,y,z) and a constraint g(x,y,z). What if their gradients are not parallel to each other at the point of their touch? May there be such situations?

Thank you.

The gradients indeed must be parallel at the extrema (assuming our functions are nice and differentiable). For example, this is explained and proved in Lecture 13 of the MIT series ( https://m.youtube.com/watch?v=15HVevXRsBA ). The proof of it begins from around 27:00 in that video.

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Re: Multi-variable systems?

Postby Boris Lagutin » Mon Aug 03, 2015 1:27 pm

Thanks a lot, a.s.h. I am sorry I suppose I asked the question obscurely. I understand that those gradients must be parallel. My question is about constraints and their gradients' relations with the gradients of the main (so to speak) function. Let's go from another side. As I understood there may be various kinds of constraints for functions. It depends on a situation which such a function describes. 1) May a constraint not to touch a function at extrema at all? 2) If the constraint doesn't touch the function at extrema how to apply Lagrange multiplier to find extrema of the function?

P.S. Let's take an example of f(P,V,T) so P - pressure, V - volume, T - temperature and a constraint PV = n*R*T then assume a situation when the constraint doesn't touch the function at its extrema (if possible in this case).
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Re: Multi-variable systems?

Postby Boris Lagutin » Mon Aug 03, 2015 8:48 pm

P.S. I try to understand how physical, geometrical and mathematical notions of constraints with respect to functions may be related. In this way I try to analyze this conception starting in different ways.
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Re: Multi-variable systems?

Postby Boris Lagutin » Thu Sep 03, 2015 12:05 am

By the way, my understanding of Gradient vector's conception is well illustrated at this picture: http://www.math.umd.edu/~petersd/241/html/ex12_01.png where N is a gradient vector as I understand. However, vectors stated as Gradient vectors and shown in the video: https://www.khanacademy.org/math/multiv ... gradient-1 are DIFFERENT because all of them are parallel to the bottom. But one can see that the gradient in the picture is not parallel to the bottom. Comparing the first diagram and the second diagram (in the video) some questions emerge, don't they? I have some thoughts about it but would like to hear others' thoughts.

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Re: Multi-variable systems?

Postby joe » Thu Sep 03, 2015 8:42 am

The first link you give is not explained. However, I guess that the surface it shows is an equipotential (e.g. a surface of constant voltage V(x,y,z) ) and the normal vector it shows is the gradient vector (e.g. grad(V), which is minus one times the electric field). I didn't watch the video (busy).

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Re: Multi-variable systems?

Postby Boris Lagutin » Thu Sep 03, 2015 9:59 am

Thank you very much, Joe.

As I understand, Gradient vector is ALWAYS perpendicular to a tangent plane at a point which this Gradient vector comes from. There are a surface in the picture and another surface in the video. One can sees that Gradient vector (N) in the picture is perpendicular to the tangent plane, however, all of Gradient vectors in the video are parallel to x,y-plane (bottom). In this way, Gradient vectors in the video cannot be perpendicular to the tangent planes at appropriate points! Is it possible that Gradient vectors are NOT perpendicular to appropriate tangent planes?

Thank you.
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Re: Multi-variable systems?

Postby joe » Thu Sep 03, 2015 11:02 am

So far as I can see, if grad(V) exists, it is always at right angles to a surface of equal V.
Perhaps the surfaces are not equipotentials. Or perhaps the author means something different when s/he says gradient vector. For instance, be aware of the difference between gradient and gradient vector.

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Re: Multi-variable systems?

Postby Boris Lagutin » Thu Sep 03, 2015 11:06 am

Thank you, Joe. However, he didn't mention such points if I didn't miss something. In my opinion it may lead to confusions.
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Re: Multi-variable systems?

Postby Boris Lagutin » Sun Sep 06, 2015 12:47 am

If F is a force vector that is a gradient vector to a curve in the case #1 how this vector may be NOT perpendicular to this curve at points which such vectors come from? Because this curve is not equipotential?

As we see, in the case #2 it's clear that the force vectors are perpendicular to the curve.

Thank you.
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Re: Multi-variable systems?

Postby joe » Sun Sep 06, 2015 7:19 am

In the sketch shown, F is obviously not the total force. (BTW not all the F vectors are at right angles to the curve)

A case where the total force on an object is at right angles to its trajectory is uniform circular motion (not the case here)

If this sketch comes from a course offered by someone else, you should ask that person what they mean.

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Re: Multi-variable systems?

Postby Boris Lagutin » Sat Sep 12, 2015 6:12 am

There are 2 regions S_1 and S_2: a portion of plane and a portion of paraboloid on the drawing. It's clear that these areas are not equal each other. Could someone explain how to figure out the following:

1) Area of S_2 by using the double integral?

2) Flux through area S_2 by using just a double integral shown on the drawing?

Thanks a lot.
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Re: Multi-variable systems?

Postby joe » Mon Sep 14, 2015 9:57 am

In the first one, dA = dx dy
so A = integral dx dy = delta x * delta y

For the other one, the question is not very clear. But if the field is F k, where F is constant and k is the unit vector in the z direction, then, from the definition of flux, the flux is just
integral F k . dA2 = integral F dA1 = F * delta x * delta y

If F is not constant, then the integral is harder.


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