kelvin56484984 said:Homework Statement
F(x,y,z)=4x i - 2y^2 j +z^2 k
S is the cylinder x^2+y^2=4, 0<=z<=3
Homework Equations
The Attempt at a Solution
[/B]
I find that the answer is 84 pi
What is the difference after if I change the equation to inequality?
For example :
x^2+y^2<=4, z=3
x^2+y^2<=4 , z=0
X^2+y^2=4, z=3
thanks!
kelvin56484984 said:Homework Statement
F(x,y,z)=4x i - 2y^2 j +z^2 k
S is the cylinder x^2+y^2=4, 0<=z<=3
Homework Equations
The Attempt at a Solution
[/B]
I find that the answer is 84 pi
What is the difference after if I change the equation to inequality?
For example :
x^2+y^2<=4, z=3
x^2+y^2<=4 , z=0
X^2+y^2=4, z=3
thanks!
Bottom: ##\vec R(x,y) = \langle x,y,0\rangle,~0\le x^2+y^2\le 4##kelvin56484984 said:Sorry, I made some mistake previously and I modified the question.
If I want to find the flux of the bottom face, top face, side surface of cylinder and the tangent surface of the cylinder,
How can I express it in inequality?
The Divergence Theorem with Inequality is a mathematical theorem that relates the flux of a vector field through a closed surface to the triple integral of the divergence of the same vector field over the enclosed volume. It is a generalization of the Divergence Theorem, which only applies to equalities.
The Divergence Theorem with Inequality is used in physics to relate the flow of a vector field through a closed surface to the sources and sinks of that vector field within the enclosed volume. This is useful in understanding fluid flow, electric and magnetic fields, and other physical phenomena.
The Divergence Theorem only applies to equalities, while the Divergence Theorem with Inequality allows for inequalities. This means that the Divergence Theorem with Inequality is a more general theorem that can be applied to a wider range of situations.
The Divergence Theorem with Inequality is derived from the Divergence Theorem using mathematical techniques such as Stoke's Theorem and the Cauchy-Schwarz inequality. The proof involves breaking the surface into smaller pieces and applying the Divergence Theorem to each piece, then taking the sum of these pieces and applying the Cauchy-Schwarz inequality to obtain the inequality.
The Divergence Theorem with Inequality has many practical applications, such as in fluid mechanics, electromagnetism, and even computer graphics. It is used to calculate flow rates, analyze electric and magnetic fields, and simulate fluid behavior, among other things. This theorem has also been applied in economics, biology, and other fields to model and analyze various systems.