A thin metal plate, located in the x y-plane, has temperature T(x, y) at the point (x, y). The l

A thin metal plate, located in the x y-plane, has...

Key Concepts

Quadratic Forms

Quadratic forms involving variables like x and y, particularly when they appear as squared terms with different coefficients, can define conic sections such as ellipses, parabolas, or hyperbolas. In the context of level curves for the given temperature function, the quadratic form in the denominator typically gives rise to elliptical shapes indicating how the temperature values distribute in space.

Sketching Level Curves

Sketching level curves involves setting the function equal to different constant values and solving for the variables to obtain the corresponding curves. This process provides a visual understanding of the function's behavior, showing curves that encapsulate the same output value, and in thermal problems, these curves are known as isotherms.

Level Curves

Level curves are the loci of points in the domain of a function of two variables where the function maintains a constant value. They are widely used in multivariable calculus to visualize functions by drawing the set of points corresponding to different constant outputs, which helps in understanding how the function behaves across its domain.

Isotherms

Isotherms are a special case of level curves used in thermal contexts. They represent curves along which the temperature is constant, making them particularly useful in studying heat distribution. In physical applications like a metal plate, isotherms help to visualize regions that share the same temperature.

Transcript

00:02 Okay, so we're given the function t of xy, and it's equal to 100 over 1 plus x squared plus 2y squared.

00:13 And so we want to know the level curves of this function, so we're going to set it equal to z.

00:19 Right now it's pretty hard to tell, but why don't we rearrange this equation into something we're more familiar with? so let's raise both sides to the negative first power.

00:43 And then, oops, this should be raised to the negative first.

00:52 Alright, and so now let's multiply through by 100 on each side and rewrite.

01:10 All right.

01:11 And then let's just get our variables alone.

01:19 So i'm going to subtract 1 to this side.

01:22 And this entire fraction is over z, so that's the same thing as, 100 minus z or z.

01:34 All right.

01:37 And so the reason we're able, by the way, to rearrange this without any worry of changing the graph is there are no discontinuities anywhere on this graph.

01:50 Our variables are strictly contributing positive values to the denominator, and the denominator is never zero.

01:59 So you can be pretty confident that this won't change.

02:02 Our graph.

02:06 All right.

02:07 And so let's just make our xy plane.

02:11 All right.

02:15 And so what is the largest? what is this actually? why did we rearrange it like this? i'll tell you what.

02:27 This is a radius.

02:28 This on this side you should be familiar with as some type of parabola, or not parabol.

02:37 Sorry, in ellipse where an ellipse has the equation a x squared plus b y squared is equal to some constant c squared well we already said that z is fixed so this is a constant and the square root of it is going to be not necessarily our radius but it's just a constant all right and so and so another thing to notice is our x is going to extend how much further than our y.

03:28 Let's assume that y is zero.

03:30 So x squared equals, let's just call this some constant c.

03:40 Let's leave one to distinguish from that formula.

03:45 So x is going to extend between negative and positive c1.

03:52 Y, on the other hand, will extend between negative and positive c1 over 2, square root of c1 over 2.

04:04 Because we have to divide through by 2 before we can take the square root...

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