How can I find this surface integral in cylindrical coordina

• John004
In summary, the problem involves calculating the flux of a vector field through a closed surface bounded by two cylinders and two planes. The vector field is defined in cylindrical polar coordinates and the unit vectors are given. The first part of the question asks to calculate the flux using a surface integral, while the second part asks to use the divergence theorem. The student is confused about how to approach the problem and asks for clarification on how to represent the unit normal vector and infinitesimal area elements. They also ask if they should take four separate integrals and sum them afterwards.
John004

Homework Statement

A vector field $\vec F$ is defined in cylindrical polar coordinates $\rho , \theta , z$ by

$\vec F = F_0(\frac{xcos (\lambda z)}{a}\hat i \ + \frac{ycos(\lambda z)}{a}\hat j \ + sin(\lambda z)\hat k) \ \equiv \frac{F_0 \rho}{a}cos(\lambda z)\hat \rho \ + F_0sin(\lambda z)\hat k$

where $\hat i$ , $\hat j$ , and $\hat k$ are the unit vectors along the cartesian axes and $\hat \rho$ is the unit vector $\frac{x}{\rho}\hat i \ + \frac{y}{\rho}\hat j$.

(a) Calculate, as a surface integral, the flux of $\vec F$ through the closed surface bounded by the cylinders $\rho = a$ and $\rho = 2a$ and the planes $z =\pm \frac{a\pi}{2}$.

(b) Evaluate the same integral using the divergence theorem.

So I am pretty lost on how to even begin part (a). I sketched a picture of the situation and am I kinda confused as to how to represent the unit normal vector, as well as the infinitesimal area elements. Am I supposed to take four separate integrals and sum them afterwards, one for the inner and outer curved surfaces, and one for the upper and lower caps?

I see that this is marked solved, so I haven't commented. I just fixed the text to make it readable. @John004: use ## instead of \$ to delimit inline Latex.

John004 said:

Homework Statement

A vector field ##\vec F## is defined in cylindrical polar coordinates ##\rho , \theta , z## by

$$\vec F = F_0(\frac{xcos (\lambda z)}{a}\hat i \ + \frac{ycos(\lambda z)}{a}\hat j \ + sin(\lambda z)\hat k) \ \equiv \frac{F_0 \rho}{a}cos(\lambda z)\hat \rho \ + F_0sin(\lambda z)\hat k$$

where ##\hat i## , ##\hat j## , and ##\hat k## are the unit vectors along the cartesian axes and ##\hat \rho## is the unit vector ##\frac{x}{\rho}\hat i \ + \frac{y}{\rho}\hat j##.

(a) Calculate, as a surface integral, the flux of ##\vec F## through the closed surface bounded by the cylinders ##\rho = a## and ##\rho = 2a## and the planes ##z =\pm \frac{a\pi}{2} ##.

(b) Evaluate the same integral using the divergence theorem.

So I am pretty lost on how to even begin part (a). I sketched a picture of the situation and am I kinda confused as to how to represent the unit normal vector, as well as the infinitesimal area elements. Am I supposed to take four separate integrals and sum them afterwards, one for the inner and outer curved surfaces, and one for the upper and lower caps?

What is a surface integral in cylindrical coordinates?

A surface integral in cylindrical coordinates is a mathematical concept that involves finding the area of a curved surface in three-dimensional space. In cylindrical coordinates, the surface is defined by a radius and an angle, rather than Cartesian coordinates x, y, and z.

Why do we use cylindrical coordinates for surface integrals?

Cylindrical coordinates are often used for surface integrals because they can simplify the calculations and make them easier to solve. Additionally, many geometrical shapes, such as cylinders and cones, are naturally described using cylindrical coordinates.

How do I convert a surface integral from Cartesian coordinates to cylindrical coordinates?

To convert a surface integral from Cartesian coordinates to cylindrical coordinates, you can use the following formulas:

For the limits of integration:
x = rcosθ, y = rsinθ, and z = z
For the surface element:
dS = rdzdrdθ

What are the advantages of using cylindrical coordinates for surface integrals?

There are several advantages to using cylindrical coordinates for surface integrals. First, they can simplify the equations and make them easier to solve. Second, they are often more intuitive for describing curved surfaces. Finally, cylindrical coordinates can be particularly useful when dealing with problems that involve rotational symmetry.

What are some common applications of cylindrical coordinates for surface integrals?

Surface integrals in cylindrical coordinates have many practical applications in science and engineering. They are often used in fields like fluid mechanics, electromagnetism, and signal processing. For example, in fluid mechanics, surface integrals are used to calculate the flow rate through a curved pipe or the force exerted by a fluid on a curved surface.

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