Applications 53–56. Ideal flow A two-dimensional vector field describes ideal flow if it has both zero curl and zero divergence on a simply connected region (excluding the origin if necessary). a. Verify that the curl and divergence of the given field is zero. b. Find a potential function φ and a stream function ψ for the field. c. Verify that φ and ψ satisfy Laplace’s equation φ x x + φ y y = ψ x x + ψ y y = 0 . 55. F = 〈 tan − 1 y x , 1 2 ln ( x 2 + y 2 ) 〉
Applications 53–56. Ideal flow A two-dimensional vector field describes ideal flow if it has both zero curl and zero divergence on a simply connected region (excluding the origin if necessary). a. Verify that the curl and divergence of the given field is zero. b. Find a potential function φ and a stream function ψ for the field. c. Verify that φ and ψ satisfy Laplace’s equation φ x x + φ y y = ψ x x + ψ y y = 0 . 55. F = 〈 tan − 1 y x , 1 2 ln ( x 2 + y 2 ) 〉
Solution Summary: The author explains the formula used to verify that the vector field F=langle mathrmtan-1yx,
53–56. Ideal flowA two-dimensional vector field describes ideal flow if it has both zero curl and zero divergence on a simply connected region (excluding the origin if necessary).
a. Verify that the curl and divergence of the given field is zero.
b. Find a potential function φ and a stream function ψ for the field.
c.Verify that φ and ψ satisfy Laplace’s equation
φ
x
x
+
φ
y
y
=
ψ
x
x
+
ψ
y
y
=
0
.
55.
F
=
〈
tan
−
1
y
x
,
1
2
ln
(
x
2
+
y
2
)
〉
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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The vector field F(x, y) = (x + y )i + 2xyj is graphed below. Use the image to
locate a point where the curl is negative, and then show that the curl is negative at
your point by computing the curl there.
10
-5
-10
-10
-5
10
APR
29
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Let (z, y, z)=
+ z ln (y + 2) be a scalar field.
Find the directional derivative of at P(2,2,-1) in the direction of the vector
12
4
Enter the exact value of your answer in the boxes below using Maple syntax.
Number
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