Logarithmic potential Consider the potential function φ ( x , y , z ) = 1 2 ln ( x 2 + y 2 + z 2 ) = ln | r | , where r = 〈 x , y, z 〉 . a. Show that the gradient field associated with ϕ is F = r | r | 2 = 〈 x , y , z 〉 x 2 + y 2 + z 2 . b. Show that ∬ S F ⋅ n d S = 4 π a , where S is the surface of a sphere of radius a centered at the origin. c. Compute div F. d. Note that F is undefined at the origin, so the Divergence Theorem does not apply directly. Evaluate the volume integral as described in Exercise 37.
Logarithmic potential Consider the potential function φ ( x , y , z ) = 1 2 ln ( x 2 + y 2 + z 2 ) = ln | r | , where r = 〈 x , y, z 〉 . a. Show that the gradient field associated with ϕ is F = r | r | 2 = 〈 x , y , z 〉 x 2 + y 2 + z 2 . b. Show that ∬ S F ⋅ n d S = 4 π a , where S is the surface of a sphere of radius a centered at the origin. c. Compute div F. d. Note that F is undefined at the origin, so the Divergence Theorem does not apply directly. Evaluate the volume integral as described in Exercise 37.
Solution Summary: The author explains the gradient field associated with phi and the vector field.
Logarithmic potential Consider the potential function
φ
(
x
,
y
,
z
)
=
1
2
ln
(
x
2
+
y
2
+
z
2
)
=
ln
|
r
|
, where r = 〈x, y, z〉.
a. Show that the gradient field associated with ϕ is
F
=
r
|
r
|
2
=
〈
x
,
y
,
z
〉
x
2
+
y
2
+
z
2
.
b. Show that
∬
S
F
⋅
n
d
S
=
4
π
a
, where S is the surface of a sphere of radius a centered at the origin.
c. Compute div F.
d. Note that F is undefined at the origin, so the Divergence Theorem does not apply directly. Evaluate the volume integral as described in Exercise 37.
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.
R.p
Plot the gradient vector field of f together with a contour map of f. Explain how they are related to each other. f (x, y) = ln(1 + x^2 + 2y^2)
Use Green's Theorem in the form of this equation to prove Green's first identity, where D and C satisfy the hypothesis of Green's Theorem and the appropriate partial derivatives of f and g exist and are continuous. (The quantity ∇g · n = Dng occurs in the line integral. This is the directional derivative in the direction of the normal vector n and is called the normal derivative of g.)
Gradient fields on curves For the potential function φ and points A, B, C, and D on the level curve φ(x, y) = 0, complete the following steps.a. Find the gradient field F = ∇φ.b. Evaluate F at the points A, B, C, and D.c. Plot the level curve φ(x, y) = 0 and the vectors F at the points A, B, C, and D.
φ(x, y) = y - 2x; A(-1, -2), B(0, 0), C(1, 2), and D(2, 4)
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