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Active Calculus - Multivariable

Nicholas Long, Mitchel T. Keller, Steve Schlicker

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Activity 13.1.2.
The plot in Figure 13.1.6 illustrates the vector field \(\vF(x,y) = y\vi -x\vj\text{.}\)
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described in detail following the image
A vector field plotted in the plane with \(x\) and \(y\) both ranging from \(-5\) to \(5\text{.}\) The vectors have a counterclockwise rotation about the origin, with vectors getting progressively longer as they get farther from the origin.
Figure 13.1.6. The vector field \(y\vi-x\vj\)
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(a)
Starting with one of the vectors near the point \((2,0)\text{,}\) sketch a curve that follows the direction of the vector field \(\vF\text{.}\) To help visualize what you are doing, it may be useful to think of the vector field as the velocity vector field for some flowing water and that you are imagining tracing the path that a tiny particle inserted into the water would follow as the water moves it around.
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(b)
Repeat the previous step for at least two other starting points not on the curve you previously sketched.
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(c)
What shape do the curves you sketched in the previous two steps form?
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(d)
Verify that \(\vF(x,y)\) is orthogonal to \(\langle x,y\rangle\text{.}\)
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(e)
Calculate the gradient of the function \(f(x,y) = x^2 + y^2\) and write a sentence comparing your result to the vector \(x\vi + y\vj\text{.}\)
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(f)
Write a sentence describing the geometric relationship between \(\vF(x,y)\) and a circle centered at the origin. What is the relationship between \(\vecmag{\vF(x,y)}\) and the radius of that circle?
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