5

Point) ShowF(z,y) = (4ry? + 2)i + (6x2y? + 2e2y)jis conservative by finding potential function f for F , and use f to compute F dr , where C is the curve given byF(...

Question

Point) ShowF(z,y) = (4ry? + 2)i + (6x2y? + 2e2y)jis conservative by finding potential function f for F , and use f to compute F dr , where C is the curve given byF(t) = 2sin" ti + 2t sin12 5tjfor 0 < t < "/2f(z,y) =IP.& =

point) Show F(z,y) = (4ry? + 2)i + (6x2y? + 2e2y)j is conservative by finding potential function f for F , and use f to compute F dr , where C is the curve given by F(t) = 2sin" ti + 2t sin12 5tj for 0 < t < "/2 f(z,y) = IP.& =



Answers

Sketch the curve represented by the vector-valued function and give the orientation of the curve. $$\mathbf{r}(t)=2 \sin t \mathbf{i}+2 \cos t \mathbf{j}+e^{-t} \mathbf{k}$$

Okay, we're going to sketch the curve represented by the vector valued function. We do have um second and tangent. So um we're gonna be putting in angles but before I do that, I want to consider that second is one over cosine and tangent is sine over cosine. Of course. I used theaters in one situation and teas and the other. I'll make sure to use betas in both since we're in terms of data for this problem. But the main idea here is that we have a denominator, right? That denominator means that we are undefined in some places. So we will be considering that as we go through this um to think about where co sign of theta equals zero. So let's go ahead and list some of those angles that we want to try out. I am putting some extra angles in here because as we stated, you know, we do know co sign up pi over two is undefined. So I did throw it there so that we would see that. Um But I also put in some other values and um so notice we will be dealing with um some square roots of three, you know square 23 is like 2.7. So you can kind of figure out that is without having a conductor calculator. Okay so let's go ahead and sketch out, notice we have X's first that are positive and then some that are negative. And then remember I only went from zero to pi so we might want to consider because there's a whole, you know I feel like there's peace here that's that I'm missing. And what I noticed that I am hyperbole to. And then I also notice that in age situation you know we do start low and end high for each of our sections and so we've used our arrows to show our direction.

Okay, we're going to sketch the curve represented by the vector valued function and then give its orientation. So first of all, looking at this um you may kind of know what this looks like. Look at the co sign T. In the I. Direction and the sign T. In the J. Direction. Without the twos, they would be a unit circle of radius one. Well now with the twos, they are just a circle with radius too. But what's happening as you go around the circle, you're also gaining height in a nice linear fashion. A nice um you know um constant rate in the Z. So I kind of know what this looks like. But let's go ahead and put some numbers in so that we can actually use our our coordinate system to do some sketching. So for my times I'm going to use things that are nice on the unit circle. So I'm using negative pi over 20 pi over two and pi. Now as I put my co sign in I know that at both negative pi over two and pi over two, that's where I'm zero and then I will be at two and negative two for my zero and pi now my sign is the opposite. Remember I have my maxes and men's at my pies over twos and my zeros at zero and pi now Z. Is actually going to be the values. And the main thing here is it's constant. So um time is going to be going out a nice constant value. Okay so to plot these I'll go ahead and get my X. Y. Z. Axis. And then I consider that if I'm going to go in the X. Direction zero so no zero and then go in the Y direction negative too. And then I'd have to go um you know down then I go to the 0.200 and then I go 02 and then up a pi over two. So I'm trying to the same amount that I went down before. I'm trying to go up now and then I need to go double that up by the time I get to the back you know where I'm going to be you know um behind that and then it just keeps swirling up at a nice um constant rate. And so what it does is it looks like a nice little spiral um or like a slinky that's been stretched out.

The problem is finding a point on the curve artiness, but you two times consigned you two times feinting into tears. He from zero to high wire hats untended alive, his parallel just lying. You took three AKs. Us. Why it's there or what? First to the computer county this is connected to has signed true times. Signe and it's normal. Lecture supplying. It's the culture into three. One serial space. The canned lies How Lelouch is fine. We have he from Don't productive and this leaf with cereal. So we have collective to three times. Find you last call sign. It's called zero. So we have It's green comes Signe Yes, coach Coastline so tender and team. It's because of y our virgins. So we have. He is twenty high over six. So what? King is cultural pie over six. So it's a point on the crew of this arm. Hide over. Six. This isn't you. Two times it's a three over to you. Two hands need to high over six Distance, three one on each. Hiring over six

Okay, we're going to sketch the curve represented by the vector value function. So we're going to be putting in some values and we're going to be putting in angles. Remember when it says co sign to the third T they're really saying do the cosine of the angle and then take the whole thing to the third power. So some of these are going to be easy to do because co sign is often zero or one. So we are going to fill in all of our zeros. Um Cosign has zeros at our negative pi over two and pi over two were sign which is ry piece has its zeros at zero and pi now the max is a means of both the X and Y piece are going to be negative two and two right? Because it depends on whether our co sign value is a positive one or a negative one. And when you take it to the third power it's still going to be one or negative one. So at zero we get that positive one. So we'll have it too. And then a negative two at pi where we'll have our negative two at negative pi or two for sign into four pi or two. So I did include power before I knew that these would only be our major ones and we're gonna have to figure out what's happening between the major ones because of the third power. But I assume because I have fractions taking something to the third power is going to make it even more of a fraction. But let's go ahead and plot what we have and then we can figure out what else we need. So you can see that there's definitely cemetery going on. Now if I do put a pi over four in the co sign or sign of pi over four is squirted to over two. So that needs to be taken to the third power and multiplied by two. So I have my two out front. The square root of two to the third power will be like a two square root of two. And then I'll just list my three twos on the bottom so I can cross out a few and I get square to to over two. Now let's 20.707 So I know it's a little bit smaller than one. So this will help me graph that point. And so once I do graph that point and I think about all my other quadrants, I know that I'm still going to get these points 707 at all my thoughts. And so it's going to have the same shape all the way around. And I wasn't surprised that um you know in a fraction and you take it to the third power. Of course it's going to get smaller. Now our direction is going to be counter clockwise as we go around this curve


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