Simple Surface Integral of a Cylinder: Homework Statement and Solution Attempt

In summary, the conversation discusses the process of integrating a vector field over the surface of a cylinder with given limits. The person is struggling with the surface integral and has attempted to solve it using divergence, but is unsure of their answer. The conversation ends with a question about where they may have made a mistake.
  • #1
cpfoxhunt
16
0

Homework Statement



This is annoying me because I am so clearly being a muppet somewhere. I need to integrate the vector field (x,-y,z).(vector)ds over the surface of a cyliner x^2 + y^2 < 4 (or equal to) and z is between 0 and 1.

The Attempt at a Solution



I have to do it both with and without divergence. With divergence, div(a) = 1, integral over a nice volume, gives me 4Pi, as O a, (integrating 2*r drd(theta)dz, with the specified limits.)

Trying to do it as a surface integral is driving me nuts. I end up integrating r^2(sin^2(x) - cos^(x) between the limits x = 0 and x = pi. as this integrates to sin(x)cos(x), this makes the entire integral zero.

So where am I going wrong?
 
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  • #2
I don't think you included the contributions from the flat ends of the cylinder, did you?
 

What is a simple surface integral?

A simple surface integral is a mathematical tool used to calculate the total flux or flow of a vector field across a two-dimensional surface. It involves integrating the dot product of the vector field and the surface's normal vector over the surface's area.

What is the difference between a simple surface integral and a double integral?

A simple surface integral is a type of double integral that specifically deals with integrating over a two-dimensional surface. In contrast, a double integral can involve integrating over a region in two-dimensional space, such as a square or a circle.

How is a simple surface integral calculated?

To calculate a simple surface integral, the surface must first be parameterized using two variables, typically denoted as u and v. The integral is then set up as the double integral of the dot product of the vector field and the surface's normal vector, with the limits of integration determined by the parameterization.

What is the physical significance of a simple surface integral?

A simple surface integral has several physical interpretations, depending on the context in which it is used. For example, in fluid dynamics, it represents the total flow of a fluid across a surface. In electromagnetism, it represents the total electric flux through a surface. In general, it can be used to calculate the amount of a physical quantity passing through a surface.

What are some real-world applications of simple surface integrals?

Simple surface integrals have many applications in physics, engineering, and other fields. They are commonly used in fluid dynamics to calculate the flow of fluids across surfaces, in electromagnetism to calculate electric and magnetic flux, and in heat transfer to calculate the rate of heat transfer across surfaces. They are also useful in computer graphics for rendering three-dimensional objects and in geographic information systems for terrain modeling.

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