- #1
calculo2718
- 25
- 1
I am trying to solve problems where I calculate work do to force along paths in cylindrical and spherical coordinates.
I can do almost by rote the problems in Cartesian: given a force ##\vec{F} = f(x,y,z)\hat{x} + g(x,y,z)\hat{y}+ h(x,y,z)\hat{z}## I can parametricize my some curve ##\gamma: \vec{r}(t) = x(t)\hat{x} + y(t)\hat{y} + z(t)\hat{z}## then take ##\int_\gamma \vec{F}\cdot\vec{r}'(t)dt ##
I know it should work similarly in cylindrical, if I have ##\vec{F}(R,\phi) = f(R,\phi)\hat{R} + g(R,\phi)\hat{\phi}##, The problem is I am having trouble thinking through the curve parametrization in cylindrical and similarly in spherical. For example, If I want to integrate along the path where ##\phi = 0##, from some radius ##r_1## to ##r_2## I would say
##\gamma_1: \vec{r}(t) = t\hat{R} + 0\hat{\phi}##
If I then want to go from along the quarter circle path from ##\phi = 0## to ##\phi = \frac{\pi}{2}## then
##\gamma_2: \vec{r}(t) = r_2\hat{R} + r_2\cdot t\hat{\phi}##
If I want to go back down to ##r_1##
##\gamma_3: \vec{r}(t) = t\hat{R} + \frac{\pi}{2}\hat{\phi}##
This is where my confusion gets in the way, the ##\gamma_2## curve makes it look like I can in general parametricize a curve in cylindrical as ##\vec{r}(t) = R(t)\hat{R} + \phi(t)R(t)\hat{\phi}## where on this curve ##R(t) = r_2## and ##\phi(t) = t##, however the ##\gamma_3## curve is not of this form and if I try to parametricize it that way(##\vec{r}(t) = t\hat{R} + \frac{\pi}{2}t\hat{\phi}##) it is incorrect(I tested it with a Force only in the ##\hat{\phi}## direction which should give me 0, but I get a non-0 integral).
How can I parametricize a general curve in cylindrical and spherical the way I have done above for cartesian? What am I missing when trying to do these problems?
In addition how can I derive from my ##\vec{r}(t)## in cylindrical the line element in cylindrical ##d\vec{r} = dR\hat{R} + rd\phi\hat{\phi}##. I feel like understand how this is done will help me see more clearly why I am confused
I can do almost by rote the problems in Cartesian: given a force ##\vec{F} = f(x,y,z)\hat{x} + g(x,y,z)\hat{y}+ h(x,y,z)\hat{z}## I can parametricize my some curve ##\gamma: \vec{r}(t) = x(t)\hat{x} + y(t)\hat{y} + z(t)\hat{z}## then take ##\int_\gamma \vec{F}\cdot\vec{r}'(t)dt ##
I know it should work similarly in cylindrical, if I have ##\vec{F}(R,\phi) = f(R,\phi)\hat{R} + g(R,\phi)\hat{\phi}##, The problem is I am having trouble thinking through the curve parametrization in cylindrical and similarly in spherical. For example, If I want to integrate along the path where ##\phi = 0##, from some radius ##r_1## to ##r_2## I would say
##\gamma_1: \vec{r}(t) = t\hat{R} + 0\hat{\phi}##
If I then want to go from along the quarter circle path from ##\phi = 0## to ##\phi = \frac{\pi}{2}## then
##\gamma_2: \vec{r}(t) = r_2\hat{R} + r_2\cdot t\hat{\phi}##
If I want to go back down to ##r_1##
##\gamma_3: \vec{r}(t) = t\hat{R} + \frac{\pi}{2}\hat{\phi}##
This is where my confusion gets in the way, the ##\gamma_2## curve makes it look like I can in general parametricize a curve in cylindrical as ##\vec{r}(t) = R(t)\hat{R} + \phi(t)R(t)\hat{\phi}## where on this curve ##R(t) = r_2## and ##\phi(t) = t##, however the ##\gamma_3## curve is not of this form and if I try to parametricize it that way(##\vec{r}(t) = t\hat{R} + \frac{\pi}{2}t\hat{\phi}##) it is incorrect(I tested it with a Force only in the ##\hat{\phi}## direction which should give me 0, but I get a non-0 integral).
How can I parametricize a general curve in cylindrical and spherical the way I have done above for cartesian? What am I missing when trying to do these problems?
In addition how can I derive from my ##\vec{r}(t)## in cylindrical the line element in cylindrical ##d\vec{r} = dR\hat{R} + rd\phi\hat{\phi}##. I feel like understand how this is done will help me see more clearly why I am confused