(a) Use Green's Theorem to prove that ∫ C f x d x + g y d y = 0 if f and g are differentiable functions and C is a simple, closed, piecewise smooth curve. (b) What does this tell you about the vector field F x , y = f x i + g y j ?
(a) Use Green's Theorem to prove that ∫ C f x d x + g y d y = 0 if f and g are differentiable functions and C is a simple, closed, piecewise smooth curve. (b) What does this tell you about the vector field F x , y = f x i + g y j ?
(a) Use Green's Theorem to prove that
∫
C
f
x
d
x
+
g
y
d
y
=
0
if f and g are differentiable functions and C is a simple, closed, piecewise smooth curve.
(b) What does this tell you about the vector field
F
x
,
y
=
f
x
i
+
g
y
j
?
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.
The vector field F is shown below in the xy-plane and looks
the same in all other horizontal planes. (In other words, F is independent
of z and its z-component is 0.)
(1) Is divF positive, negative, or zero? Explain.
(2) Determine whether curl F = 0. If not, in which direction does curlF
point?
3. Let f(x, y) = sin x + sin y. (NOTE: You may use software for any part
of this problem.)
(a) Plot a contour map of f.
(b) Find the gradient Vf.
(c) Plot the gradient vector field Vf.
(d) Explain how the contour map and the gradient vector field are
related.
(e) Plot the flow lines of Vf.
(f) Explain how the flow lines and the vector field are related.
(g) Explain how the flow lines of Vf and the contour map are related.
The figure shows a vector field F and three paths from P (-3,0) to
Q= (3,0). The top and bottom paths T and B comprise a circle, and the middle
path M is a line segment. Determine whether the following quantities are positive,
negative, or zero, or answer true or false. Be sure you can explain your answers.
(Click on graph
to enlarge)
(a)
F dr is ?
(b)
F- dř is ?
F. dr is ?
(c)
F-dr is 2
()
(e) ?
v True or False:
F- dr
() ?
v True or False F is a gradient field.
University Calculus: Early Transcendentals (4th Edition)
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