In a lake the water is moving and at each point in the lake there is a velocity vector so, we have a well-defined velocity field V at any point of the lake. Let us consider a thinkable curve C in the lake. Then the line integral V.T ds from the velocity field V with respect to the arc-length measure over the curve (path) C sums (adds up) the tangent components to the curve C of the velocity field V along the points on the curve C. In this sense, the line integral measures how much of the water (the fluid) is aligned with the curve C. In other words, the above line integral measures, indicates how much of the water (the fluid) tends to circulate around the curve C. Hence, we define the circulation r done by the velocity field V around the curve C as r = V.T ds In physics, circulation is the line integral of a vector field around a close curve. In fluid dynamics the field is the fluid velocity field. In electrodynamics, it can be the electric or the magnetic field. Suppose that a curve C is traced by the vector function r. Then s= V.T ds V.dr C Problem: -y x + 1 i + (x + 1)²+4y² Find the circulation done by the velocity field V(x,y) = (x + 1)? + 4y2 Around the positively oriented square with the vertices (2,0), (0,2), (-2,0) and (0,-2)
In a lake the water is moving and at each point in the lake there is a velocity vector so, we have a well-defined velocity field V at any point of the lake. Let us consider a thinkable curve C in the lake. Then the line integral V.T ds from the velocity field V with respect to the arc-length measure over the curve (path) C sums (adds up) the tangent components to the curve C of the velocity field V along the points on the curve C. In this sense, the line integral measures how much of the water (the fluid) is aligned with the curve C. In other words, the above line integral measures, indicates how much of the water (the fluid) tends to circulate around the curve C. Hence, we define the circulation r done by the velocity field V around the curve C as r = V.T ds In physics, circulation is the line integral of a vector field around a close curve. In fluid dynamics the field is the fluid velocity field. In electrodynamics, it can be the electric or the magnetic field. Suppose that a curve C is traced by the vector function r. Then s= V.T ds V.dr C Problem: -y x + 1 i + (x + 1)²+4y² Find the circulation done by the velocity field V(x,y) = (x + 1)? + 4y2 Around the positively oriented square with the vertices (2,0), (0,2), (-2,0) and (0,-2)
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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Transcribed Image Text:In a lake the water is moving and at each point in the lake there is a velocity vector so, we have
a well-defined velocity field V at any point of the lake. Let us consider a thinkable curve C in the
S.
lake. Then the line integral
V.T ds from the velocity field V with respect to the arc-length
C
measure over the curve (path) C sums (adds up) the tangent components to the curve C of the
velocity field V along the points on the curve C. In this sense, the line integral measures how
much of the water (the fluid) is aligned with the curve C. In other words, the above line integral
measures, indicates how much of the water (the fluid) tends to circulate around the curve C.
Hence, we define the circulation s done by the velocity field V around the curve C as r=
V.T ds
C
In physics, circulation is the line integral of a vector field around a close curve. In fluid dynamics
the field is the fluid velocity field. In electrodynamics, it can be the electric or the magnetic field.
Suppose that a curve C is traced by the vector function r. Then r =
V.T ds =
V.dr
C
Problem:
-y
x + 1
Find the circulation done by the velocity field V(x,y) =
i +
(x + 1)? + 4y2
(x + 1)²+4y²J
Around the positively oriented square with the vertices (2,0), (0,2), (-2,0) and (0,-2)
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