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B. Consider now the vector field F() = (22 + yP)i + 2k _ where 2 = ci +yj + zk. Express F(E) in a spherical coordinate system: Namely: F() = F(r,0 ,o)7 (z) + Folr,9...

Question

B. Consider now the vector field F() = (22 + yP)i + 2k _ where 2 = ci +yj + zk. Express F(E) in a spherical coordinate system: Namely: F() = F(r,0 ,o)7 (z) + Folr,9,0)0 () + Fo(r,9,0)o(w). You may use only the following definitions: x = r sin 0 cos $ y = r sin 0 sin $ 2 = r COS 0and:r(z) = sin 0 COS $ & + sin 0 sin $ j + cos 0 k 0 (2) cos 0 COS 0 i + cos 0 sin $ j sin 0 k $(z) = _ sin $ i + COS 0 j

B. Consider now the vector field F() = (22 + yP)i + 2k _ where 2 = ci +yj + zk. Express F(E) in a spherical coordinate system: Namely: F() = F(r,0 ,o)7 (z) + Folr,9,0)0 () + Fo(r,9,0)o(w). You may use only the following definitions: x = r sin 0 cos $ y = r sin 0 sin $ 2 = r COS 0 and: r(z) = sin 0 COS $ & + sin 0 sin $ j + cos 0 k 0 (2) cos 0 COS 0 i + cos 0 sin $ j sin 0 k $(z) = _ sin $ i + COS 0 j



Answers

Give a formula $\mathbf{F}=M(x, y) \mathbf{i}+N(x, y) \mathbf{j}$ for the vector field in the plane that has the properties that $\mathbf{F}=0$ at $(0,0)$ and that at any other point $(a, b), \mathbf{F}$ is tangent to the circle $x^{2}+y^{2}=a^{2}+b^{2}$ and points in the clockwise direction with magnitude $|\mathbf{F}|=$ $\sqrt{a^{2}+b^{2}}$

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Hello there. So for this exercise we need to sketch the some of these back to back to fields but for given values of A. Okay, so let's remember a little bit that the plot for the vector field F. Is given by old vectors that are pointing away from the origin. Okay. And the length of course of these vectors are going to increase as if we were far from the origin. Okay. So if you put a negative value in front of this vector field, the only thing that we're going to do is with the vectors instead of pointing away from the origin, they're going to point uh towards the origin. Okay, so that's the idea now. What is the vector field G. Well, the vector field G. Is all the vectors that are pointed around the the yes, they're pointing around the origin. Okay. Like this way they were going around the origin. So they have to describe. Okay, so what is going to happen if A is equal to zero If A is negative or is or if a is positive. So I'm going to show you with more interactive sketch. Okay, okay. So here we have the plot. So this is for a we drove here. This is for for a equals to zero. Okay, when A is equal to zero, we only have this plot. That is all the vectors are circulating around the origin. So that's why you have you can see like kind of vortex like uh vector field. But all the all the factors are pointing around the origin. Never point forward or request if you're something to this vector field, the interaction of vector field that is pointing away from the origin. What you're going to do is taking these vectors and rotating a little bit. That means that instead of pointing here this vector to left, it's going to point diagonal. This is going to be pointed like that. So you're going to rotate a little bit these vector fields and that's what you're going to observe in these. Okay. So if we take positive values of A. Because remember that our vector field F. We have A F. Plus um was G. So if A. Is positive then we are taking the positive values of A. Okay, so that's why we're taking here. So look if A is close to zero, then we have just the normal the vector field. But when we start to modify the body of a, you can observe the value. These victories start to rotate. Okay. To that to the right. Okay. You can see they start to point away from there. And if you start to take negative values of your emerging the relations with the vector field F. And the vector field F. Is the one that points away from the origin. Okay? So if you change Now you take a negative then you're back. You're going to point towards the origin. So that's why you're going to observe here. That the relation of these back these factors here are going to start, they're going to start to 2.2 towards the origin. Okay. And you can observe it right here. If we start to take negative values at this point we have taken start to take negative values for a. And you can see that all these vectors start point towards the origin and uh they preserve this structure of rotating around the origin. But now they are pointing to the origin positive values. They point away from the origin negative values. They point towards the origin. And for zero value you have just the inspector field that is circulating around the arch. But

Hello there. Okay, so for this exercise we have these two vector fields F. And G. And let's remember that the vector field F. Is the vector field where all these factors are pointing away from the origin. Okay. And the vector G. Vector field G. Is the vector field where all the vectors are pointed in circular direction with respect to the origin. Okay. So they are around the origin. So now we need to see what happened with the interaction of these two vector fields. What happened if we sum them up? But we have a small perimeter that modifies how strong his interaction with the vector field F. So we need to see what happened when B is equal to zero. I think that that is trivial. We're only, yeah, going to have only the influence of the vector field F. So, we're going to have this scenario at what happened if B is positive or if B is negative. So you can observe that for be positive. You're going to have this scenario here where these vessels are going to point in club, concert club wind direction. And if we will change the sign with for negative values of B, they're going to be in the opposite direction, clockwise direction. Okay, so let me show you this with a better, more interactive sketch. Okay, so here we have our sketch here, I have the value of B. You can observe the B is equal to zero, and we have the expected vector field will wear all these vectors are pointing away from the origin. What happened if we start to take positive values of B? We're going to some the interaction with the vector field that rotate in a counterclockwise direction. And you can observe how these vectors start to rotate and they start to kind of circulate like pointing in a counterclockwise direction, but they preserved that structure from the vector field F. That all these factors you can observe. They start to point away from the region if you're some interaction, the only thing that you're with the vector field F. The only thing that you're doing is rotating them. So they have this kind of vortex like structure. If you take for example the constitute you can observe more than than the the vector field G start to dominate because they have a factor B. B. It represents. How strong is the interaction with the vector field? We take here a really big value of B. Then we're going to observe just the structure of the vector field. Now what happens if we change the value of the for negative values? Well the rotation is going to change to the opposite side. So now You can observe that for -2. This vector field start to, the vectors start to rotate in the club wise direction, but they preserved that the structure from the vector field F where all these these vectors are pointing away from the Earth and that they are bigger as they are far from the origin. Okay.


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