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 . 54. F = ( x 3 – 3 xy 2 , y 3 – 3 x 2 y )
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 . 54. F = ( x 3 – 3 xy 2 , y 3 – 3 x 2 y )
Solution Summary: The author explains that the vector field F=langle x3-3y2 has zero curl and zero divergence.
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
.
54. F = (x3 – 3xy2, y3 – 3x2y)
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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