Match equations "a" though "d" with vector fields "A" though "D", as shown on the attachment.

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Expert Answers
ishpiro eNotes educator| Certified Educator

In order to match these two-dimensional vector field with their equations, try to predict how the x- and y- components of each field will vary with the values of x and y, and then check which of the graphs shows the field with expected behavior.

For example, for the field a), the x-component is zero everywhere. This means that the field vectors will only have y-component, or be vertical, everywhere. This is case on the graph (D). Notice also that the length of the field vector gets larger as the absolute value of x gets larger, which is consistent with the y-component being equal to the square of x. 

For the field b), the x-component is x - y. This means the x-component is zero whenever x = y, which is on the line with the equation y = x, or bisector of the angle between the positive x- and y-axes. The field vectors on this line should be vertical. This is the case on the graph (C). Another feature that confirms that b) corresponds to (C) is that the y-component of the field vectors is positive and gets larger as x is positive and gets larger. This is consistent with the fact that y-component equals to x.

For the field c), the x-component of the field vectors is expected to be positive when x is positive (and negative when x is negative). However, y-component should be negative when y is positive, and vice versa. This is true for the graph (B).

Finally, for the field d) (which of course has to match (A) because this is the only one left), both x-and y- components of the field vectors are positive in the first quadrant, as expected. In the second quadrant, the y-component is negative but the x-component is still positive, which is again consistent. Notice also that the field vectors at the line y = x are directed at 45 degrees with the horizontal, because the x-and y-components of the field have to be equal there.

To summarize:

a) - (D)

b) - (C)

c) - (B)

d) - (A)