Multi-variable systems?

[joe]

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.

[Boris Lagutin]

Ok. I agree with you

[Boris Lagutin]

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.

[a.s.h.]

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.

[Boris Lagutin]

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).

[Boris Lagutin]

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.

[Boris Lagutin]

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.

Thank you.

[joe]

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).

Joe

[Boris Lagutin]

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.

[joe]

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.

[Boris Lagutin]

Thank you, Joe. However, he didn't mention such points if I didn't miss something. In my opinion it may lead to confusions.

[Boris Lagutin]

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.

[joe]

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.

[Boris Lagutin]

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.

[joe]

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.