Can someone help me with this please.
A vector field in cylindrical polar co-ordinates is given by
V = R Rˆ + R 3 s i n ( φ ) c o s ( φ ) φˆ + 3 z kˆ
where Rˆ, φˆ, kˆ are the appropriate unit vectors. Translate this vector field into the
Cartesian x, y, z co-ordinate system.
What is the first step??
Could you please guide me through the steps.
Thanks.
Not sure what the R 3 means, but the transformation is just the usual polar-Cartesian we all know and love:
x = r cosφ
y = r sinφ
z = z
yeahhh i get that but how do you translate theta in terms of cos or sine?
i know i need to sub 'unit R^' with cosφ i + sinφ j and z with 'unit K^' with z k.
How about 'unit φ^' ?
What do i need to sub phi with?
I look else where it says sub it with -sinφ i + cosφ j.
But how? and why? can someone explain it to me?
Thankssss
The unit vectors are just the vectors perpendicular to the cylindrical surface.
r^ points radially outward
φ^ points tangent to the planar circle
z^ points upward
To convert to Cartesian vectors,
r^ = cosφ i + sinφ j
φ^ = -sinφ i + cosφ j
z^ = z^
You have V = rrˆ + r^3 sinφ cosφ φˆ + 3z kˆ
Now, the position of any point on the curve is
V.(r^,φ^,z^)
= r(cosφ i + sinφ j) + (r^3 sinφ cosφ)(-sinφ i + cosφ j) + 3z k
= (rcosφ-r^3 sin^2φ cosφ)i + (rsinφ+r^3sinφcos^2φ)j + 3z k
= (x-xy^2)i + (y+x^2y)k + 3zk
Thanks you, Steve. It does make a bit of sense to me know.
But just to clarify, since φ is the points tangent to the planer circle,
thus, φ is the derivative of the points x, y, z?
To translate the vector field from cylindrical polar coordinates (R, φ, z) to Cartesian coordinates (x, y, z), we can use the following relationships:
x = R * cos(φ)
y = R * sin(φ)
The Cartesian unit vectors i, j, and k are aligned with the x, y, and z-axis, respectively.
To get the first step, let's break down the given vector field into its components:
V = R Rˆ + R^3 sin(φ) cos(φ) φˆ + 3z kˆ
The first component, R Rˆ, represents the radial direction and can be translated directly into Cartesian coordinates as follows:
Vx = R Rˆ * cos(φ) = R * cos(φ)
Vy = R Rˆ * sin(φ) = R * sin(φ)
The second component, R^3 sin(φ) cos(φ) φˆ, represents motion in the azimuthal direction. To translate it into Cartesian coordinates, we can use the relationships:
φˆ = -sin(φ) i + cos(φ) j
So, we have:
Vx' = R^3 sin(φ) cos(φ) * (-sin(φ)) = -R^3 sin^2(φ) cos(φ)
Vy' = R^3 sin(φ) cos(φ) * cos(φ) = R^3 sin(φ) cos^2(φ)
The third component, 3z kˆ, represents motion in the z-direction. As kˆ is aligned with the z-axis in Cartesian coordinates, we have:
Vz = 3z
Putting it all together, the translations of the vector field into Cartesian coordinates are:
Vx = R * cos(φ) - R^3 sin^2(φ) cos(φ)
Vy = R * sin(φ) + R^3 sin(φ) cos^2(φ)
Vz = 3z
These equations give the components of the vector field in Cartesian coordinates (x, y, z).