 We're now going to take a look at general conduction analysis. And so in order to do this, what we're going to do, we're going to start off with Fourier's law, which if you recall is the law that applies for analyzing conductive heat transfer. So if you recall Fourier's law, which is the conduction rate equation, what we said was that the heat flux, and now we looked at it for one dimension, so let's say it's in the x direction, is related to the thermal conductivity, the area through which that heat is flowing, and then the gradient of temperature in the dimension that we're looking at. So in this case the x direction. Now this equation is observed. It is not a derived equation, so it's observed from experiments. So it's what we call a phenomenological equation. So this equation, phenomenological, is derived from experimentation and from observations. Now the equation that we've shown here is for a one-dimensional case, and this equation can be extended into multiple dimensions. And so we can extend it to three dimensions, and let's write that out. And so what we can do, we can write the heat flux as a vector. And so here we can see we have the x component in the i direction, the y component in the j, and the z components in the k direction. And we can also relate this to Fourier's law. And so let's do that in this line. And what we'll do, we'll pull the minus k a, so the thermal conductivity multiplied by the projected area out of the brackets. And then what is left on the inside is just going to be the derivative with respect to the dimension. But instead of being an ordinary differential, what we'll now have is we're going to have partial differentials, because we're looking at a case where the temperature could be a function of multiple variables. So looking at this, we can rewrite this as the heat flux, which is a vector, is minus k the thermal conductivity times an area. And looking at what we have in the brackets here, if you review your vector, vector mathematics, you'll recall that one of the vector operators was this is the gradient operator. And it's the gradient of the temperature, temperature T, which is a scalar. So here the gradient and the gradient, if you recall, from your math courses is defined in the following way. And we would have whatever it is that we're operating on in brackets here. So that's the gradient operator. And that's essentially Fourier's law in three dimensions then for all three spatial dimensions. Now, what we can do, we can illustrate what is going on graphically. And so let's take a look at what that looks like. And we'll look at this in 2D. So let's imagine we have some object and we'll write it out as being x and y. And if we could go in and measure the temperature at every point in that object, and then plot up what we call isotherm. So those would be lines of constant temperature. We would get something like this. So if those were our lines of constant temperature, and if we were to take this field and we were to compute the gradient of this field, we would get lines that we call gradient lines. So I'll sketch those on and the gradient lines are going to be perpendicular to the isotherm lines. And if we were to take a particular point within this space, so let's take this point right here, and we'll call that point xy. And we were to evaluate the gradient at that point. Remember the gradient is going to give us a vector. So it's going to have direction. So the gradient is going to be in this direction here. And essentially what it is representing is the direction of the maximum change in temperature. So that would be the gradient of temperature. And if I keep drawing a line, okay, so we get that. Now, if you recall, and we know just from physical experience that heat always flows from a hot point to a colder point. And consequently, what we're seeing here is the gradient is showing us the direction in which the heat is flowing. And it's going to go from a region of high temperature to low temperature. And so the gradient is just simply showing us the direction that the heat is flowing in. And it happens to be that where you get the maximum change in temperature with position. And that's what the gradient operator is showing. So the heat is flowing in the direction of maximum temperature decrease. Now, if we want to evaluate the heat flux, remember the heat flux is a vector, because we had q is minus k, a gradient of t. And t is a scalar. This is going to give us a vector when we take the gradient operator of the scalar temperature field. So if you want to be able to evaluate the heat flux q, what this is telling us, usually you'll know your thermal conductivity, but we need to be able to evaluate the temperature field in order to get the gradient of the temperature field. So that is one of the main goals within heat transfer analysis, depending upon the problem. But usually for conduction, what we're after is to determine the temperature distribution within an object. So in order to proceed with this and start doing calculations, we need a way to be able to find out what the temperature is within an object. Okay. So we need a way to be able to find the temperature field. And so that's what we're going to do in the next segment. We're going to derive an equation. It's going to take a little bit of time. It's the heat diffusion equation. But this is the most general equation for conduction analysis. And it enables you to determine the temperature field in an object provided you know a lot of information about that object, mainly the boundary conditions, what the material is on the inside. But nonetheless, what we'll do will derive the heat diffusion equation. And with that equation, you can determine the temperature field. And from that, you can get heat flux and determine the flow of heat throughout an object.