What’s wrong? Consider the rotation field F = ( − y , x ) x 2 + y 2 . a. Verify that the two-dimensional curl of F is zero, which suggests that the double integral in the circulation form of Green’s Theorem is zero. b. Use a line integral to verify that the circulation on the unit circle of the vector field is 2 π . c. Explain why the results of parts (a) and (b) do not agree.
What’s wrong? Consider the rotation field F = ( − y , x ) x 2 + y 2 . a. Verify that the two-dimensional curl of F is zero, which suggests that the double integral in the circulation form of Green’s Theorem is zero. b. Use a line integral to verify that the circulation on the unit circle of the vector field is 2 π . c. Explain why the results of parts (a) and (b) do not agree.
Solution Summary: The author explains that the two dimensional curl of the vector field is zero.
What’s wrong? Consider the rotation field
F
=
(
−
y
,
x
)
x
2
+
y
2
.
a. Verify that the two-dimensional curl of F is zero, which suggests that the double integral in the circulation form of Green’s Theorem is zero.
b. Use a line integral to verify that the circulation on the unit circle of the vector field is 2π.
c. Explain why the results of parts (a) and (b) do not agree.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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