Preview Activity 13.6.1.
In this preview activity, we will look at several two-dimensional vector fields and try to assess when the vector field has increased or decreased in strength over a given region. We begin with graphs of the three vector fields, \(\vF\text{,}\) \(\vG\text{,}\) and \(\vH\text{.}\) Parts a, b, and c ask you to answer the same three questions about the vector field and square illustrated in each of the figures. Part d asks you to think further about the third vector field.
A vector field with vectors radiating from the origin. The length of vectors increases as distance from the origin increases. There is a square in the first quadrant with sides parallel to the coordinate axes. Centered in the square is a point labeled \(P_1\text{,}\) which appears to lie on the line \(y=x\text{.}\)
A vector field with vectors below the \(x\)-axis pointing primarily up near the axis and primarily to the right farther away from the axis. Above the \(x\)-axis, vectors point primarily to the right. Vector magnitudes are shortest near the positive \(y\)-axis. There is a square in the second quadrant with sides parallel to the coordinate axes. Centered in the square is a point labeled \(P_2\text{,}\) which appears to lie on the line \(y=-x\text{.}\)
A vector field in which vectors circulate around the origin as if tangent vectors to concentric circles centered at the origin. Vector magnitude increases as distance from the origin increases. There is a square in the first quadrant with sides parallel to the coordinate axes. Centered in the square is a point labeled \(P_3\text{.}\) There is a point labeled \(P_4\) in the second quadrant that is the reflection of \(P_3\) across the \(y\)-axis. In the third quadrant, there is a point \(P_5\) that is closer to the \(y\)-axis than to the \(x\)-axis. In the fourth quadrant, there is a point labeled \(P_6\) that appears to be on the line \(y=-x\) but is farther from the origin than \(P_3\) or \(P_4\text{.}\)
(a)
For each of the vector fields \(\vF\text{,}\) \(\vG\text{,}\) and \(\vH\) and the square centered on \(P_1\text{,}\) \(P_2\text{,}\) and \(P_3\) (respectively), which statement do you think best applies?
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More of the vector field is going into the square than going out.
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Less of the vector field is going into the square than going out.
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The same amount of the vector field is going into the square as is going out.
(b)
For each of the vector fields \(\vF\text{,}\) \(\vG\text{,}\) and \(\vH\) (and corresponding square), does your answer to part a suggest that the vector field is being created, destroyed, or is unchanging in strength inside the square? Write a sentence to explain your thinking for each vector field.
