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Hello, everyone. This is Tom from everystepcalculus.com. A problem dealing with curl. And whether it’s conservative or not. So let’s do it. Index 8 to get to my menu. Scroll down to curl. They’re all alphabetical. Conservative. And we have the choice of PQR or MNP system. I’m going to use number two MNP system this time. And we’re going to add the function. Alpha before you anything into these entry lines, here. 2 times Y Alpha X squared plus 2 times Y times Z. And Alpha Y squared. I always show you’ve entered so you can change it if you want. I say it’s okay. This is the cross product system for doing to curl. Those are partial derivatives. And we compute each one. This is the I cartesian system. I, J, and K, etc. These turn out to be 00. Let’s see. They’re zeros and this one is not zero. Here’s the answer for the curl. Now go into the menu. Now we want to choose whether it’s conservative. So I’m going to do number one. And so here we have the the calculations and the curl again. Does this equal zero. Say no. Because the curl does not to zero, it is not conservative. So that’s the system. Try to do the curl and try to remember the cross product system of doing those calculations. I never could memorize that that kinda stuff. I have much more important things to memorize. Alright, have a good one.

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Hello, everyone. Tom from everystepcalculus.com and everystepphysics.com. With regard to a curl, there are 2 notating systems. One, PQR and MNP. Why they chose those letters, I have no idea, in math. But that’s the way it is. Some professors use one and some professors use the other. Sometimes they use the Cartesian system which is I, J, and K, also. So I’m going to do an M, N, P system right now. Just throw in arbitrary functions and show you. Index 8 to get to my menu. We’re going to go down here to curls. We’ll go down to M, N, P. I like to, I’m going to just show you the definition of a curl, first. Curl is a vector with a magnitude (length) and direction. It represents the circulation, rotation, twist, or turning force per unit area of a field. Now that’s the best definition I can come up with because I asked that to a lot of times in calculus. I said, what the heck is this, what are you talking about? And even this, I hesitate to believe that it has any value other than some mathematical nonsense that they throw around in calculus to do these problems. A vector field is a three dimensional blob and many small vectors pointing in different directions together forming a pattern of definite form. Curl equals Twist divided by Area. A curl of F (x,y,z) is equal to the, here’s the M N P system and here’s the P Q R system defined within the Cartesian coordinates I J K. And if you’re to use the vector notation, you have these arrows on the side. M, N, P and you have arrows on the P, Q, R system. M, a function (x, y, z) in i direction. N a function (x, y, z) in j direction, and P a function (x,y,z) in k direction. Or used also is P, Q, R, etc. So now let’s do an example. We’re going to choose number 2, MNP. You have to press Alpha before you enter anything into these entry lines, here. Alpha 2 times Y squared times Z. Alpha 5 times X times Y times Z. Now these functions, I’m just dreaming up so. It’s probably going to go off the calculator. The answers. Because the screen is so small. It doesn’t go off the calculator screen on the TI-92 plus but it doesn’t on the Titanium. And then I know that you’re doing homework or experimenting because anytime an equation goes off the, think of it. The rest of your class is scored by partial credit and the class curve. Who else in the class could solve a deep problem in calculus. So will never be on a test because it’s just too hard. You would take too much time. Even if you could solve it, it would take too much time, too much paperwork, there would be no time for the rest of the problems on the test. And I have to program for tests only because of the space of the Titanium for memory use and just sanity. There’s ten million derivations of calculus derivatives and integrals. In each one of these, I have to do on a piece of paper, every one of them, to find the correct step by step process. Once I get done with that, I say it’s the best system for solving a problem but it might not be the system that your professor uses on the blackboard so you want to keep that in mind, too. Here’s the second one for the P. I’m going to put in, let’s put in Y cubed times z squared times X. oh, I didn’t press Alpha first. So we need to go back and change it. So alpha 2 times X squared times Z. Alpha 3 times 5 times X times Y squared. Alpha Z cubed. Now they seem to be correct. I’m going to say okay. Choose number 1. Here’s the matrix system for solving the curl and it’s really a cross product system. I don’t know who can remember this. I can’t, I couldn’t. That’s the reason I programmed it. Why try to memorize something when you can actually program it and have it for the rest of your life? So here’s the answers to what’s going on. With partials with respect to certain other partials. Oh the answer turned out to be not too bad, here. So now we can compute it at a point. Choose number 3. Enter the points, let’s just enter some. Alpha 1, alpha minus 5, alpha 9. Again, I show you what you’ve entered. And you substitute. These would be parentheses on your paper. These quotation marks have to be done in the programming, I can’t do anything different. And it’s substituting 1 for any X or minus 5 for any Y or 9 for any Z in the answer here. Here’s a Y right here, minus 5 but you put parentheses around that because you’re substituting. And the answer turns out to be 377. Pretty neat, huh? Have a good one.

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Hello, everyone. Tom from everystepcalculus.com and every stepphysics.com. Problem on Curl Divergence. Are they diverges or not and what is the answer of that. So let’s do it. Index 8 to get to my menu. Go down to curl. Don’t do the M,N,P system. Let’s do the P,Q,R system
right now. Depending on what your professor likes or does. Alpha to press anything in here. You have to press Alpha first. We’re going to put in 2 times Y and then Alpha X squared plus 2 times Z times Y, and then alpha Y squared. I always show you what you’ve entered,
you can change it if you want. I say it’s okay. Compute the curl first. Then we want to choose number 2. To the different formula. This is the system it’s not a crossproduct situation like it was in the curl. So each one is done partial of P with respect to partial of X. It turns out to be zero. The derivative of this is zero. Partial of Q with respect to Y turns out to be 2z. Partial of R partial with respect to Z turns out to be 0. So the answer is 2z. Now you want to compute it at a point. We can do that too, you can press one here, we can go to the and enter our point. Let’s enter alpha 1, alpha 5 and alpha 6. I show you again what you’ve entered, you can change it if you want. And these quotations marks are really parentheses for you. You’re substituting 6 for Z so you can put parentheses around the six on your test or homework.The answer is 12. Pretty neat, huh? Have a good one.

Raw Transcript

Hello, everyone. Tom from everystepcalculus.com and everystepphysics.com. I’m going to do a video of a curl at a point. And let’s do it. Index 8 to get to my menu. Scroll down to the Curl, curl at a point. We’re going to do it with the M, N, P system. Number 2. Alpha, you have to press Alpha before you enter anything into these entry lines, in my programs. Z squared times Y. Alpha 3 times X times Y. Alpha Z cubed times X. I always show you what you’ve entered. You can change it if you want. I say it’s okay. Here’s the formula for the curl. Cross product stuff that’s what we’re doing. And the cross product equals this type of answers here. And the answer is this right here. So we can go back to compute it to a point. Press number 3. Alpha 2. Alpha minus 6. Alpha 9. Show you what that is, too so you can change it. And the variables go in here, in this line here and here’s the answer 936. Let’s do some other variables here. Alpha 2, alpha 3, alpha 4. 47 is the answer. In these quotations here, that’s where you put parentheses because you’re substituting the xyz values for answer of the curl. And you put parentheses around there but I can’t do that with the programming. Pretty neat, huh? Go to my site, subscribe to me to see other videos. Have a good one.