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.
Stream function Recall that if the vector field F = ⟨ƒ, g⟩ is source free (zero divergence), then a stream function ψ exists such that ƒ = ψy and g = -ψx.a. Verify that the given vector field has zero divergence.b. Integrate the relations ƒ = ψy and g = -ψx to find a stream function for the field.
F = ⟨y2, x2⟩
Stream function Recall that if the vector field F = ⟨ƒ, g⟩ is source free (zero divergence), then a stream function ψ exists such that ƒ = ψy and g = -ψx.a. Verify that the given vector field has zero divergence.b. Integrate the relations ƒ = ψy and g = -ψx to find a stream function for the field.
F = ⟨-e-x sin y, e-x cos y⟩
Deteremine if values are positive, negative, or zero from given vector field
F*dr - from where C is a line from (5,0) to (0,5)
F*nds where C is a circle centered around (0,0)
F*dr where C is a circle centered around (0,0)
Single Variable Calculus: Early Transcendentals (2nd Edition) - Standalone book
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