# partial differential equations best video lectures

set dy to be zero. extremely clear at the end of class yesterday. This constant k actually is called the heat conductivity. applies to each particle. how they somehow mix over time and so on. We would like to get rid of x because it is this dependent variable. Pretty much the only thing to remember about them is that df over ds, in the direction of some unit vector u, is just the gradient f dot product with u. That is pretty much all we know about them. this situation where y is held constant and so on. Who prefers that one? Now we are in the same situation. Expect a problem about reading, a contour plot. So, we have to keep our minds open and look at various possibilities. Well, we don't have actually four independent variables. A partial differential equation is an equation that involves the partial derivatives of a function. Who prefers this one? Here we use it by writing dg equals zero. It is the top and the bottom. One of them is to find the minimum of a maximum of a function when the variables are not independent, and that is the method of Lagrange multipliers. That is a critical point. This goes away and becomes zero. Now I want partial h over partial x to be zero. But, of course, we are in a special case. It tells you how well the heat flows through the material that you are looking at. So that will be minus fx g sub z over g sub x plus f sub z times dz. Where did that go? But in a few weeks we will, actually see a derivation of where this equation comes from. In fact, the really mysterious part of this is the one here. And that is an approximation for partial derivative. And it sometimes it is very. You will see. and something about constrained partial derivatives. questions like what is the sine of a partial derivative. We are replacing the graph by its tangent plane. And that causes f to change at that rate. If you take the differential of f and you divide it by dz in this situation where y is held constant and so on, you get exactly this chain rule up there. We need to know -- --, directional derivatives. The other method is using the chain rule. » is just the gradient f dot product with u. We will be doing qualitative questions like what is the sine of a partial derivative. Well, how quickly they do that is precisely partial x over partial u, partial y over partial u, partial z over partial u. And, when we plug in the formulas for f and g, well, we are left with three equations involving the four, What is wrong? That chain rule up there is this guy, df, divided by dz with y held constant. Another topic that we solved just yesterday is constrained, partial derivatives. To take this into account means, that if we vary one variable while keeping another one fixed. gradient of g. There is a new variable here. Basically, what causes f to change is that I am changing x, y and z by small amounts and how sensitive f is to each. You can just use the version that I have up there as a, template to see what is going on, but I am going to explain it, That is the most mechanical and mindless way of writing down the, chain rule. And we have learned how to package partial derivatives into. How does it change because of y? There was partial f over partial x times this guy. Sorry, depends on y and z and z, what is the rate of change of f with respect to z in this, Let me start with the one with differentials that hopefully you, kind of understood yesterday, but if not here is a second, we will try to express df in terms of dz in this particular. Now, how to solve partial differential equations is not a topic for this class. So, we plan to make this course in two parts – 20 hours each. In our new terminology this is partial x over partial z with y, held constant. Somewhat more sophisticated but equally good is Introduction to Partial Differential Equations with Applications by E. C. Zachmanoglou and Dale W. Thoe.It's a bit more rigorous, but it covers a great deal more, including the geometry of PDE's in R^3 and many of the basic equations of mathematical physics. And then we add the effects, good-old chain rule. That is basically all we need to know about it. That is actually minus 100 exactly. Knowledge is your reward. And we have seen how to use the gradient vector or the partial derivatives to derive various things such as approximation formulas. Now, y might change, so the rate of change of y would be the rate of change of y with respect to z holding y constant. But another reason is that, really, you need partial derivatives to do physics and to understand much of the world that is around you because a lot of things actually are governed by what is called partial differentiation equations. Which points on the level curve satisfy that property? And we used the second, derivative to see that this critical point is a local, for the minimum of a function, well, it is not at a critical, boundary of the domain, you know, the range of values, that we are going to consider. The reason for that is basically physics of how heat is transported between particles in fluid, or actually any medium. And so, for example. but we can also keep using the chain rule. Flash and JavaScript are required for this feature. Topics covered: Partial differential equations; review. Now what is next on my list of topics? Use OCW to guide your own life-long learning, or to teach others. If you want, this is the rate of change of x with respect to z when we keep y constant.