Divergence of radial fields Calculate the divergence of the following radial fields. Express the result in terms of the position vector r and its length | r |. Check for agreement with Theorem 17.10 20. F = 〈 x , y , z 〉 ( x 2 + y 2 + z 2 ) = r | r | 2
Divergence of radial fields Calculate the divergence of the following radial fields. Express the result in terms of the position vector r and its length | r |. Check for agreement with Theorem 17.10 20. F = 〈 x , y , z 〉 ( x 2 + y 2 + z 2 ) = r | r | 2
Solution Summary: The author analyzes the divergence of the radial vector field with the use of Theorem 17.10.
Divergence of radial fields Calculate the divergence of the following radial fields. Express the result in terms of the position vectorr and its length |r|. Check for agreement with Theorem 17.10
20.
F
=
〈
x
,
y
,
z
〉
(
x
2
+
y
2
+
z
2
)
=
r
|
r
|
2
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.
Properties of div and curl Prove the following properties of thedivergence and curl. Assume F and G are differentiable vectorfields and c is a real number.a. ∇ ⋅ (F + G) = ∇ ⋅ F + ∇ ⋅ Gb. ∇ x (F + G) = (∇ x F) + (∇ x G)c. ∇ ⋅ (cF) = c(∇ ⋅ F)d. ∇ x (cF) = c(∇ ⋅ F)
1. You are given a vector function A = Îx,
(a) Sketch this vector function in the x-y coordinate plane.
(b) Without doing the math, look at your sketch, and explain in words why this vector
function does or does not have a divergence.
(c) Calculate the divergence of this vector function. Is the answer a scalar or a vector?
University Calculus: Early Transcendentals (4th Edition)
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