Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The vector field F = 〈3 x 2 , 1〉 is a gradient field for both φ 1 ( x , y ) = x 3 + y and φ 2 ( x , y ) = y + x 3 + 100 . b. The vector field F = 〈 y , x 〉 x 2 + y 2 is constant in direction and magnitude on the unit circle. c. The vector field F = 〈 y , x 〉 x 2 + y 2 is neither a radial field nor a rotation field.
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The vector field F = 〈3 x 2 , 1〉 is a gradient field for both φ 1 ( x , y ) = x 3 + y and φ 2 ( x , y ) = y + x 3 + 100 . b. The vector field F = 〈 y , x 〉 x 2 + y 2 is constant in direction and magnitude on the unit circle. c. The vector field F = 〈 y , x 〉 x 2 + y 2 is neither a radial field nor a rotation field.
Solution Summary: The author evaluates whether the statement "The vector field F=langle 3x2,1rangle" is true or not.
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The vector field F = 〈3x2, 1〉 is a gradient field for both
φ
1
(
x
,
y
)
=
x
3
+
y
and
φ
2
(
x
,
y
)
=
y
+
x
3
+
100
.
b. The vector field
F
=
〈
y
,
x
〉
x
2
+
y
2
is constant in direction and magnitude on the unit circle.
c. The vector field
F
=
〈
y
,
x
〉
x
2
+
y
2
is neither a radial field nor a rotation field.
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.
Single Variable Calculus: Early Transcendentals (2nd Edition) - Standalone book
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