Streamlines and equipotential lines Assume that on ¡ 2 , the vector field F = { f , g ) has a potential function φ such that f = φ x and g = φ y , and it has a stream function ψ such that f = ψ y and g = – ψ x . Show that the equipotential curves (level curves of φ ) and the streamlines (level curves of ψ) are everywhere orthogonal.
Streamlines and equipotential lines Assume that on ¡ 2 , the vector field F = { f , g ) has a potential function φ such that f = φ x and g = φ y , and it has a stream function ψ such that f = ψ y and g = – ψ x . Show that the equipotential curves (level curves of φ ) and the streamlines (level curves of ψ) are everywhere orthogonal.
Streamlines and equipotential lines Assume that on ¡2, the vector field F = {f, g) has a potential function φ such that f = φx and g = φy, and it has a stream function ψ such that f = ψy and g = –ψx. Show that the equipotential curves (level curves of φ) and the streamlines (level curves of ψ) are everywhere orthogonal.
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.
A vector field representing wind speeds is given by v = ry³ i + 3x³y³j.
A walker can take one of two paths between points A(0,0) and B(1,1). Determine which path has more work done by the field?
a)
The first path follows y = x4. Find the work done by the field following this path.
For path c1,
v • dr =(
)dr
Sa v• dr =
Note: Show solution to integral as answer rounded to 2 decimal places.
b)
The second path follows y = x. Find the work done by the field following this path.
For path c2,
v • dr =(
)dx
S,v• dr
Note: Show solution to integral as answer rounded to 2 decimal places.
c)
Which path should you take if you want to put in less effort?
O Path on curve 1
O Path on curve 2
O Either path, effort is the same.
Sketch the vector field F(x,y) → xi + 3yi
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Sketch and describe the vector field F (x, y) = (-y,2x)
Chapter 17 Solutions
Calculus: Early Transcendentals and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition) (Briggs, Cochran, Gillett & Schulz, Calculus Series)
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