Need help with some vector calculus problems

[formulajoe]

f = 2xy in the x dir, (x^2 - z^2) in the y dir, -3xz^2 in the z dir.
particle travles from (0,0,0) to (2,1,3)

im supposed to find the line integral from (0,0,0) to (2,1,3).
if i make z = 3/2 x and y = 1/2 x and substitute those into the integral of F dot dl i end up with an answer of -27.66667. the way I am doing it makes sense from the things I've seen on the net. but the book gives an answer of -39.5. ??

also, I've got another problem, my electro-magnetics prof didnt get a chance to do any examples on this stuff. D = 2 Ro z^2 in the Ro dir, and Ro cos^2 theta in the z dir.
find closed surface integral of D dot Ds.
0<Ro<5
-1<z<1
0<theta<2pi

ive got no idea on this one. is it basically the same thing as a line integral or do i do something different?

 

[HallsofIvy]

I have no idea what you mean by "f = 2xy in the x dir, (x^2 - z^2) in the y dir, -3xz^2 in the z dir". Do you possibly mean 2xy dx+ (x²- z²)dy- 3xz²dz ? Oh, wait, you mean f is a vector: f= (2xy)i+ (x²- z²)j- 3xz²k.
Of course its the same thing: fds= 2xy dx+ (x²- z²)dy- 3xz²dz
To integrate it from (0, 0, 0) to (2, 1, 3), let x= 2t, y= t, z= 3t so that dx= 2dt,
dy= dt, dz= 3dt and substitute. The integral will be for t= 0 to 1.

I'm a little put off by using "Ro" instead of r in cylindrical coordinates. That's normally reserved for polar coordinates.

the figure "0< Ro< 5, -1< z< 1, 0< theta< 2p is the cylinder with radius 5, extending from z= -1 to 1. It's surface (dS) really has three parts: the top, a disk of radius 5 with z= 1 (and dS= rdrd&theta;), the bottom, a disk of radius 5 with z= -1 (and dS= -rdr&theta;), and the "lateral side" with dS= rdrd&theta;.

 

[M98Ranger]

First of all, it's great that you're seeking help with these vector calculus problems. They can be tricky, but with practice and guidance, you can master them.

For the first problem, it seems like you have the right idea in terms of setting up the line integral. However, I would recommend double-checking your calculations and making sure you're using the correct limits of integration. Also, it's always a good idea to double-check your work with the answer provided in the book. Sometimes, there may be a mistake in the book or a different approach to solving the problem. It's always good to have multiple ways of solving a problem to confirm your answer.

For the second problem, finding the closed surface integral of D dot Ds, it is a different type of integral than a line integral. It is called a surface integral, and it involves integrating over a two-dimensional surface instead of a one-dimensional line. In this case, you would need to use the formula for a surface integral and plug in the given values for the limits of integration. It's important to pay attention to the orientation of the surface and the direction of the normal vector. If you're still unsure, I would recommend seeking help from your professor or a tutor.

Overall, the key to mastering vector calculus problems is practice and understanding the concepts. Don't be discouraged if you don't get the right answer at first, keep working at it and seek help when needed. Good luck!