So, let's prove del T over del x, del T over del y, del T over del z, is a vector in a different way. We just proved it using the dot product operation, and the rationale that if the three numbers fulfill the dot product operation, then it should be a vector. So, we just discussed that we shall show that the components of any three numbers should transform, just the same way the components of a known vector, something like R, which is position vector do under rotation of the coordinate system. So, you may recall the equations to describe the change in the coordinates when you rotate the frame in the counterclockwise direction by angle Theta. In that case, the new coordinate system, which is x prime and y prime in this case, will be described like this, x prime equals x cosine Theta plus y sine Theta. Y prime equals minus x sine Theta plus y cosine Theta, and because we have rotated this frame about the axis z, the z will not change. So, it will be invariant. So, z prime will be equal to z, okay? If we do some transformation, we can also express x and y in terms of x prime and y prime using these formula. Looking at the picture, and using the geometry that you learned from maths, you will be able to understand why these equations are written in this particular way. Now, let's take a look at the position here which is P_1, and also let's take a look at P_2, which is in the neighborhood of P_1. Let's assume for simplicity that we are moving the point only along the x direction. So, the P_2 will have a new coordinate, which is x plus Delta x, comma y, comma z. So, y position, z position will be the same, only x position will be different by Delta x. So, in a prime system, which is the frame that has been rotated counterclockwise by an angle Theta, P_1 will be x prime, y prime, and z prime. P_2 in this way, you will see when you rotate it, right? You will add y component to that system. So, we will have x prime plus Delta x prime, comma y prime plus Delta y prime, and z prime, okay? Let's take a look at how the temperature difference will be expressed in a different way. So, for the prime system, temperature change, Delta T will be linear combination of the change along x prime axis and y prime axis. So, this will be the equation you want to use. If you look at the picture closely, you will understand that this will be Rho T over Rho x prime cosine Theta times Delta x because Delta x prime equals Delta x cosine Theta from this picture, right? Delta y prime equals minus Delta x sine Theta, from this picture. So, you can replace those two by this one, right? If you compare two equations above, we see that Rho T over Rho x, which will be Rho T over Rho x Delta x, will be equal to Rho T over Rho x prime cosine Theta minus Rho t over Rho y prime sine Theta, right? So, let's take a look at how x prime and y prime, which is the position vector transform under the rotation. So, you see they are one-to-one corresponding to each other. You see cosine Theta minus sine Theta, which is cosine Theta minus sine Theta times the x prime coordinate, and y prime coordinate, which is x prime coordinate of gradient, and y prime coordinate of gradient. So, in this case, we can prove del T is definitely a vector field derived from the scalar field. So, that's the end of the proof for three numbers that we just discussed. Either using dot product or see if the three numbers transform in the same way as a known vector such as position vector change. So, the next concept that we will discuss together will be the operator del. So, operator equation without a scalar field is described here. So, we just learned that del T over del x, which is the temperature change along the x axis of the original frame, is equal to Rho T over Round x prime cosine Theta, minus Rho T over Rho y prime sine Theta, which is the gradient of temperature in x prime system frame. If we remove all the functions in front of this operator, then this is what is left, okay? So, vector operator is similar to vector, but it is different from a vector in the sense that it is hungry for something to differentiate. So, it needs a function after this operator to be complete. Okay, but for convenience, we will think this vector operator as a vector, most of the vector equations we can use, right? Of course, we have some warnings, okay? So, del is an operator, and it means nothing by itself as we just mentioned. We need something after this, and del falls most of the ordinary vector algebra, which means we can use most of the vector equations. But, some exception exists because of the difference. As you can see, the order is important. In case of vector, if you multiply a vector with a scalar, it doesn't matter which way you put it. The sequence doesn't matter. But, for the operator, as you can see here, that's a big difference, okay? So, going back to our original equation where we just learned the temperature difference Delta t equals del T dot del R, you see again del T has the direction of the steepest uphill in slope in T, okay? So, Melody, can you tell us the meaning of the del T in this equation? Right, so like it says del T is the direction in the steepest uphill slope- Exactly. - of T. Yeah, and again now we understand we can use del as an operator, like a vector and T is a scalar field. Together, they form a vector, which has the direction of the maximum steepest uphill in the slope T, okay. Now, let's take a look at the operations with del. So, we will now combine del with a vector. We have learned to multiply vectors, we have two ways to do that; one is dot product, the other one is cross product. So, if we do dot product operations, del dot h, then by definition it is del sub x times h sub x plus del sub y h sub y plus del sub z h sub z, like an ordinary vector equations. If we use the partial differential equations, this is what you will get, right? For the prime system again, del prime dot h will follow the same rule, and we know that because it results in a scalar, this should be the same with del dot h, and that proves that this is a scalar field. Okay, so we think of h as a physical quantity that depends on position, space, and not strictly as a mathematical function of three variables. Okay. Then we will think about what it means. The meaning of this dot product operation with del. So, delta h is called divergence of h and we will prove in the later chapters that it really describes the flux of a field at a given point per unit volume. So for now, let's just take a note that delta h is divergence of h that is related to the flux of the vector field. Another operations, the cross-product, del cross h is called curl h, or curl of h and this is related to the circulation of a vector field at a given point per unit area. Again we will prove it, but for now let's take a note. Mathematically you can see we just use the normal vector algebra to understand the mathematical relationship between the component of each vector and the position and differential equations. So, if we revisit our Maxwell's equations and now rewrite them in vector equations, this is what you will get. So, this is neater than the forms in the previous lectures using integration and this is a point function. So, you can plot it in a space by using these equations. So the first equation delta E, is about the characteristic of E field one of which is flux and this really shows you the flux of the electric field at a given point, which is equal to the excess charge density at the point of interest, divided by the permittivity of vacuum. In most cases, as we discussed because the electric force is such a huge force, most of the time you will find the row is equal to zero. Which means you have electrical neutrality. Even you have presence of positive charges and negative charges, most of the time this will be zero. But, at a very small scale as we'll discuss for the salt or lattices of small scale molecules, this will vary as a functional position. In that case you will be able to calculate the divergence of electric field. The second equation is about circulation of electric field, which depends on the change of magnetic field. So probably you remember the Lenz law, where you have magnets underneath and if you shake the magnets up and down, you're changing the magnetic flux through this ring and because of that you're making a alternating current. Indeed that is what this is about. So, if you have change of magnetic flux, you will have circulation of electric field. However, in most cases, if you don't have this change of magnetic flux, then this term will be zero, which means there will be no circulation of electric field. Because most of the time, electric field is a gradient function of electrostatic potential and a gradient of a scalar field has no rotation. The third equation, this here, is the characteristic of magnetic field which states there is no such a thing as magnetic dipole as we discussed. So, the divergence of B will always be zero, and this is always true, up to now. The third equation is about circulation of magnetic field and we know from the Benjamin Franklin's experiment, current carrying wire will have circulation of magnetic field and that is taken care of by this term, where you have the current density at a given point. Even for capacitor where there is no direct current, if there is a change of electric field, then it will also cause circulation of magnetic field. Combining these two, we will learn that it will create electromagnetic wave. So in order to understand what we just learned about operative del and Maxwell equations, we're going to revisit the differential equation of heat flow. So Melody, can you explain to us, the students, the meaning of the equation here for the heat flow in a slab. All right. So at the top of the page we have the pretty standard equation for heat flow. So we have the thermal energy equal to a constant multiplied by temperature difference and then we have the area over the distance. However if we can conceptualize this on a [inaudible] scale, we can also quantify the change in heat flow. So then if you look here, we have the delta j, multiplied by the difference in those quantities. We talked about before how delta J divided by delta A is equal to the heat flux over here and then we see delta T over delta S, which is like those partial differentials we talked about. So then we can change that into incorporating all directions and then we get our del T here or- Exactly. So, if I recap what Melodie just told us, we can use the well known equation of heat flow which is similar to the current density equation in electricity. Where kappa is the thermal conductivity and you have the temperature difference which is the driving force for the heat flow, and geometry factor here. If we think about a very small volume and space, we can put delta in front of all the parameters that we just discussed, which will give a forum that is written here as minus kappa del T. Kappa is the thermal conductivity minus del T, will be the steepest downhill direction of the temperature. So that's the heat wants to flow, right? The steepest downhill direction of the temperature change. If you look at this equation, this is exactly what we just learned about, is the gradient of temperature and heat flow is linearly proportional to the gradient of the temperature. So, we just discussed about the first derivative of vector field or scalar field. So, to recap, we have discussed about gradient of temperature, in the previous slide we have related that with heat flow, we have discussed about divergence of electric field which is delt.E.product with the first derivative. We also have discussed about curl of electric field. So, these will all be the first derivative of scalar or vector field, we have three types. Now we are moving to the second order derivatives or second derivatives, which will give us more information about the field. But of course the contribution itself might be smaller than the first derivative. If we think about all possible combinations using the dot and cross and del that we have learned, we will come up with five different ways of expressing it, and we will go through these step-by-step. Okay. Here the T represents I think a scalar field and H is a vector. Exactly. So T could be temperature, it could be something else, but it is a scalar field. H could be heat flow, but something else, it's the vector field. So, we're going to first start with the first two equation that we saw in the list. The curl of a gradient of a scalar field. Curl of a gradient of a scalar field. We will first prove that curl of a gradient of a scalar field is always zero for any scalar function. Let's try to look at the equations. So, we learn the vector operator still, can use the same normal vector algebra. So, we can just replace del by A, any arbitrary vector and see what happens. So, A cross A times T, is always zero, because if you rearrange the functions here, you see you can have parenthesis with A cross A. We learned from the cross-product that A cross A is always zero. But mathematically you can understand that, maybe physically you may not able to understand why this is the case. Also if you use the formula that is from the definition of the curl and the gradient of a scalar field, you can also see from the relationship that we can replace it and they have the same value that it will be always zero, right? But these are all mathematical proof. So, we want to understand this equation in different way, in fiscal sense. So Melody, can you tell us about the physical meaning of this equation? Sure. If you think about this conceptually, a gradient is continuous. So, for example, if you have something in a loop like a staircase, and the staircase is continuously going up, it's impossible for this loop to be continuous. Actually, the staircase would look something like this, a spiral. Yeah. Exactly. So, that's why if you take the curl of the gradient it's zero because you can't physically realize this concept. Exactly. Thank you. That's a very neat explanation. So, as Melody explained to us, the gradient function is like a staircase, where you have increase of a function or a decrease of a function. That kind of function cannot complete a loop, i.e, it cannot have a circulation. So, that's how we can visualize the proof that we just discussed. Now, the second equation is about the.product and cross product of vector field. As you can see, this is divergence of a curl of h. Again, if you replace Del by ordinary vector A and h by B, we will also know from the vector algebra that A dot A cross B is always zero. Why? Because A cross B is a vector perpendicular to both A and B, and if you do.product between two vectors that are perpendicular, it is zero. So, we know that mathematically. However, again, we want to visualize this in a physical sense. So, I'm going to ask Melody again. Can you tell us? All right. So, if you can think of the divergence of something like a charge, for example. Here's to the right divergence of the field going out from the charge. However, if you have something circulating in a loop, there is no such divergence like that going from the loop and that's why it's zero. That's perfect. So, as you can see from what Melody just draw. If you have a circulation of a vector field at any point, you will have inflows and outflows that are matched to each other. So, you will have no net accumulation or depletion of the field. So, therefore, divergence will always be zero. So, based on those two equations, we can derive two useful theorems. The first theorem says, "If the curl A is zero, as you can see here. If Del cross A is zero, then vector A is always the gradient of something." There is some scalar field Psi such that a is equal to gradient of Psi. So, here we can see the theorem in the box. Indeed, if you revisit the Maxwell's equation and if we think about the case for electrostatic case, the second equation where you see Deltas E will become always zero because there is no change in magnetic flux as a function of time. Because of this equation, we can understand electric field will be a function of electrostatic potential, which match with this theorem. Theorem two tells us if you come across a vector field D for which divergence D is zero, then you can conclude that D is the curl of some vector field C. In a box you can see here, if Del.D equals zero, there's vector C such that D equals Del cross C. One of the example is, again, the third equation of Maxwell's equations Del.B equals zero. Therefore, B can be described as a function of the cross product of the curl of magnetic potential A. So, this types of things will be the specific examples that explain these two theorems. So, let's move on to the third part, third list of the second derivative. Here, we're going to introduce a terminology called Laplacian, which is used in many engineering fields like Navier Stokes theorems or in electrostatic world where you have Poisson's equations. It is called Laplacian and is written Del square. This is a scalar operator. We will go through first mathematical evolution, where Del.Del T, which means the divergence of gradient of temperature will be the linear combination of second partial derivative of temperature with respect to each axis. As you can see, we can take out the Del operator separately, and Del.Del is really Del squared. That's why we call this Del square T. So, Del square, which is Laplacian will be defined by Round square over Round x squared plus Round square over Round y square plus Round square over Round z square. We learn that from vector operation the.product will always result in scalar. But, as an exception here, the Laplacian can be operated on a vector as well. In that case, each Laplacian will form the components of the vector field. So, in this case, heat flow h if we do a Laplacian, x components Laplacian, y components Laplacian, and z components Laplacian will form the vector field. So, Melody, can you tell the students what Laplacian means. Unfortunately, I can't. Okay. So, probably it's a bit complicated concept. So, I'm going to give you a specific example, Fick's Second Law. Where the diffusivity times Laplacian of concentrations is equal to the change of concentration at that point. So, Laplacian here really means the curvature of your concentration in this space. The curvature of the concentrations of field will determine whether you will have out floor or inflow. Whether this will be the source of the atoms or sink of the atom. So, Laplacian in this way, you can visualize it as kind of a curvature field. So, let's move on to the fourth equation, force equation, curl of curl, curl of curl where you have Del cross Del cross h. So, you have circulation of circulation. So, it's getting very complicated. So, we will borrow some of the equations that we just discussed, A cross B cross B equals backup rule. If you replace A and B by Del, you will have this kind of equation. But, you know this is in the wrong order. So, you are going to change it. So, h can go to the right side. Once you do that, you will understand that it is combination of the gradient of the divergence and the Laplacian of the vector field. So, this form is used for electromagnetic wave, and in case of electromagnetic wave because electric neutrality is fulfilled at the scale where we think about wave, the first term you see is zero. So, we will only think about the curvature function, which is the Laplacian of h vector. Now, so let's summarize the second derivatives that we just discussed. So, we thought about five different possibilities, and we have gone through each of them step-by-step, and we have discussed the physical meaning of each equation. So, you know that Laplacian of temperature will be a scalar field, and the curl of a gradient of temperature is always zero, and the Del.h, which is divergence of heat flow, the gradient of a diverse of heat flow is a vector field, but the physical meaning is not important here. So, we will not use this a lot of time. In fact for electromagnetic wave, this will always be zero. This operation, Del cross Del cross h will be very important when they learn electromagnetic wave. So, usually, and you can see we can also have a vector field for Laplacian, which is an exceptional case for the.product operation. But, then, you may ask us, why don't we see Del cross Del among those possibilities? So, Melody, why is that the case? That is a good question. Because it's zero. Yes. So, that's correct. So, Del cross Del is always zero. Why? This is like a vector algebra, where you have the same vector and the same vector you will see if you do the cross-product will have always zero. So, some of the pitfalls when we use the vector algebra for the operators. So, be careful when applying our knowledge of ordinary vector algebra to the algebra of the operator Del. So, Del Psi, Del Phi cross product will not be zero. It might seem like it should be zero because if you replace this Del by A, it is like A Psi cross A Phi, which is always zero. But, in this case because you are doing the operation on a different function, they will give different direction, and because they will give different direction, this will not be zero. So, that's one thing. The other thing is the rules are simple and nice when we use rectangular coordinates, like cartesian coordinates. However, it gets complicated if we change to cylindrical one like r, Theta, z, z, or spherical coordinates, Rho, Theta, and Phi. So, we shall express all our vector fields in terms of their x, y, and z components when we write our vector differential equations out in components. So, with that, we'll wrap up our lecture here and hope to see you again. Thank you very much. Bye-bye.