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 . 53. F = ( e x cos y , – e x sin 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 . 53. F = ( e x cos y , – e x sin y )
Solution Summary: The author explains the formula used to verify the vector field F=langle exmathrmcosy.
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
.
53. F = (ex cos y, –ex sin y)
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.
Which vector field F has graph
y
1. F(*, у) %3
2i+xj
2. F(х, у)
(2х + 2)і + yj
3. F(х, у)
(х + у)і + (х —у)j
4. F(x, у) 3
(г — 9)i+ (х+у) j
у)i + (х+у)j
-
5. F (#, у) 3D уi+ cos a j
The vector field below contains two equilibrium solutions One equilibrium solution is shown as an orange line. What is the y
value of the other equilibrium solution? Do not include "y =" in your answer.
-6
6
Thomas' Calculus: Early Transcendentals (14th Edition)
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