Let r = x i + y j + z k, let r = r , let f be a differentiable function of one variable, and let F(r) = f ( r ) r . (a) Use the chain rule and Exercise 41(b) to show that ∇ f ( r ) = f ' ( r ) r r (b) Use the result in part (a) and Exercises 35 and 42 (a) to show that div F = 3 f ( r ) + r f ' ( r ) .
Let r = x i + y j + z k, let r = r , let f be a differentiable function of one variable, and let F(r) = f ( r ) r . (a) Use the chain rule and Exercise 41(b) to show that ∇ f ( r ) = f ' ( r ) r r (b) Use the result in part (a) and Exercises 35 and 42 (a) to show that div F = 3 f ( r ) + r f ' ( r ) .
Let
r
=
x
i
+
y
j
+
z
k,
let
r
=
r
,
let
f
be a differentiable function of one variable, and let
F(r)
=
f
(
r
)
r
.
(a) Use the chain rule and Exercise 41(b) to show that
∇
f
(
r
)
=
f
'
(
r
)
r
r
(b) Use the result in part (a) and Exercises 35 and 42 (a) to show that div
F
=
3
f
(
r
)
+
r
f
'
(
r
)
.
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.
Let f (x, y) and g(x, y) be functions of two variables withthe property that ∂ f/∂x= ∂g/∂x and ∂ f/∂y= ∂g/∂y for every point(x, y) ∈ R2. What is the relationship between f and g?
Find and sketch the domain of the function
f(x,y)= exy/ x-y2
University Calculus: Early Transcendentals (4th Edition)
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.