We have seen that all vector fields of the form F = ∇ g satisfy the equation curt F = 0 and that all vector fields of the form F = curl G satisfy the equation div F = 0 (assuming continuity of the appropriate partial derivatives). This suggests the question: are there any equations that all functions of the form f = div G must satisfy? Shoes that the answer to this question is “No” by proving that every continuous function f on ℝ 3 is the divergence of sonic vector field. [ Hint: Let G ( x , y , z ) = ⟨ g ( x , y , z ), 0, 0⟩, where g ( x , y , z ) = ∫ 0 x f ( t , y , z ) d t .]
We have seen that all vector fields of the form F = ∇ g satisfy the equation curt F = 0 and that all vector fields of the form F = curl G satisfy the equation div F = 0 (assuming continuity of the appropriate partial derivatives). This suggests the question: are there any equations that all functions of the form f = div G must satisfy? Shoes that the answer to this question is “No” by proving that every continuous function f on ℝ 3 is the divergence of sonic vector field. [ Hint: Let G ( x , y , z ) = ⟨ g ( x , y , z ), 0, 0⟩, where g ( x , y , z ) = ∫ 0 x f ( t , y , z ) d t .]
Solution Summary: The author explains that every continuous function f on R3 is the divergence of some vector field.
We have seen that all vector fields of the form F = ∇g satisfy the equation curt F = 0 and that all vector fields of the form F = curl G satisfy the equation div F = 0 (assuming continuity of the appropriate partial derivatives). This suggests the question: are there any equations that all functions of the form f = div G must satisfy? Shoes that the answer to this question is “No” by proving that every continuous function f on ℝ3 is the divergence of sonic vector field.
[Hint: Let G(x, y, z) = ⟨g(x, y, z), 0, 0⟩, where
g
(
x
,
y
,
z
)
=
∫
0
x
f
(
t
,
y
,
z
)
d
t
.]
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.
What does it mean for the differentiability of a function if only one of the Cauchy-Reimann equations (Ux = Vy and Vx = -Uy) holds?
Show that t, e^t, and sin(t) are linearly independent.
Which of the following functions are linear? Justify your answer in a comprehensible manner. In the case of a linear
function, determine the basis of the image and the core and also the associated dimensions in a comprehensible
manner.
(i) fi: R R, (a, b,c)"()
(ii) fa : R - R. (:) )
(ii) f; : R → R', (:) )
(iv) fa : R → R', (a, b,c) (1,1, 1)"
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.
RELATIONS-DOMAIN, RANGE AND CO-DOMAIN (RELATIONS AND FUNCTIONS CBSE/ ISC MATHS); Author: Neha Agrawal Mathematically Inclined;https://www.youtube.com/watch?v=u4IQh46VoU4;License: Standard YouTube License, CC-BY