Forest UCM NLM Ch1 CoordSys - Revision history

Foretony: /* Velocity vector in Spherical coordinates */

2021-08-27T20:23:56Z

Velocity vector in Spherical coordinates

2021-08-25T22:19:50Z

Velocity in Polar Coordinates

2021-08-25T22:19:22Z

Acceleration in Polar Coordinates

2021-08-25T22:18:48Z

Velocity in Polar Coordinates

2014-08-27T15:52:08Z

Velocity vector in Spherical coordinates

2014-08-27T15:50:39Z

Velocity vector in Spherical coordinates

2014-08-27T15:49:47Z

Velocity vector in Spherical coordinates

2014-08-27T15:49:30Z

Velocity vector in Spherical coordinates

2014-08-27T15:31:44Z

Velocity vector in Spherical coordinates

2014-08-27T15:29:03Z

Velocity vector in Spherical coordinates

@@ Line 202: / Line 202: @@
 :<math>\hat{\phi} = \frac{\frac{\partial \vec{r}}{\partial \phi}}{\left | \frac{\partial \vec{r}}{\partial \phi}\right | } =- \sin \phi \hat{i} + \cos \phi \hat{j}= \frac{\hat {z} \times \hat {r} }{\sin \theta}</math>
-:<math>\hat{\theta} = \hat{\phi} \times \hat {r} =\cos \phi \cos \theta \hat{i} + \sin \phi \cos \theta  \hat{j} - \sin \theta \hat{k}</math>
+:<math>\hat{\theta} = -\hat{\phi} \times \hat {r} =\cos \phi \cos \theta \hat{i} + \sin \phi \cos \theta  \hat{j} - \sin \theta \hat{k}</math>
 The derivative of the above unit vectors are

@@ Line 66: / Line 66: @@
 :<math>\frac{ d \hat{r}}{dt} = \frac{d \phi}{dt} \hat{\phi}</math>
-Thus for circular motion at a constraint radius we get the familiar expression
+Thus for circular motion at a constant radius we get the familiar expression
 :<math>\vec{v} = \lim_{\Delta t \rightarrow 0} \frac{\vec{r}(t_2)-\vec{r}(t_1)}{\Delta t}= \lim_{\Delta t \rightarrow 0} \frac{r\left(  \hat{r}(t_2) - \hat{r}(t_1)\right)}{\Delta t} = r \frac{\Delta \phi}{\Delta t} \hat{\phi} = r \omega \hat{\phi}</math>

@@ Line 135: / Line 135: @@
 Cast the unit vector in terms of cartesian coordinates and take the derivative.
-: <math> \hat{\phi} = \sin \phi \hat{i} - \cos \phi \hat{j} </math>
+: <math> \hat{\phi} = -\sin \phi \hat{i} + \cos \phi \hat{j} </math>
 :<math>\hat \dot{\phi} = \frac{d \hat{\phi}}{d \phi} \frac{d \phi}{d t} =\left( -\cos \phi \hat{i} - \sin \phi \hat{j} \right ) \dot{\phi}</math>

@@ Line 96: / Line 96: @@
 :<math>\hat r =   \cos \phi \hat{i} + \sin \phi \hat{j}</math>
-:<math>\hat \dot{r} = \frac{d \hat{r}}{d \phi} \frac{d \phi}{d t} =\left( \sin \phi \hat{i} - \cos \phi \hat{j} \right ) \dot{\phi}</math>
+:<math>\hat \dot{r} = \frac{d \hat{r}}{d \phi} \frac{d \phi}{d t} =\left( -\sin \phi \hat{i} + \cos \phi \hat{j} \right ) \dot{\phi}</math>
 :: <math>= \left ( \hat{\phi} \right ) \dot{\phi}  </math>

@@ Line 198: / Line 198: @@
 another way to determine the unit vector derivative is to cast them in terms of cartesian coordinate.
-:<math>\hat{r} = \frac{\vec{r}}{|\vec{r}|} =\cos \phi \sin \theta \hat{i} + \sin \phi \sin \theta  \hat{j} + \cos \theta \hat{k}</math>
+:<math>\hat{r} = \frac{\vec{r}}{|\vec{r}|} =\cos \phi \sin \theta \hat{i} + \sin \phi \sin \theta  \hat{j} + \cos \theta \hat{k}= \frac{\frac{\partial \vec{r}}{\partial r}}{\left | \frac{\partial \vec{r}}{\partial r}\right | } </math>
 :<math>\hat{\phi} = \frac{\frac{\partial \vec{r}}{\partial \phi}}{\left | \frac{\partial \vec{r}}{\partial \phi}\right | } =- \sin \phi \hat{i} + \cos \phi \hat{j}= \frac{\hat {z} \times \hat {r} }{\sin \theta}</math>

@@ Line 200: / Line 200: @@
 :<math>\hat{r} = \frac{\vec{r}}{|\vec{r}|} =\cos \phi \sin \theta \hat{i} + \sin \phi \sin \theta  \hat{j} + \cos \theta \hat{k}</math>
-:<math>\hat{\phi} = \frac{\frac{\partial \vec{r}}{\partial r}}{\left | \frac{\partial \vec{r}}{\partial r}\right | } = \frac{\hat {z} \times \hat {r} }{\sin \theta}=- \sin \phi \hat{i} + \cos \phi \hat{j}</math>
+:<math>\hat{\phi} = \frac{\frac{\partial \vec{r}}{\partial \phi}}{\left | \frac{\partial \vec{r}}{\partial \phi}\right | } = \frac{\hat {z} \times \hat {r} }{\sin \theta}=- \sin \phi \hat{i} + \cos \phi \hat{j}</math>
 :<math>\hat{\theta} = \hat{\phi} \times \hat {r} =\cos \phi \cos \theta \hat{i} + \sin \phi \cos \theta  \hat{j} - \sin \theta \hat{k}</math>

@@ Line 200: / Line 200: @@
 :<math>\hat{r} = \frac{\vec{r}}{|\vec{r}|} =\cos \phi \sin \theta \hat{i} + \sin \phi \sin \theta  \hat{j} + \cos \theta \hat{k}</math>
-:<math>\hat{\phi} = \frac{\hat {z} \times \hat {r} }{\sin \theta}=- \sin \phi \hat{i} + \cos \phi \hat{k}</math>
+:<math>\hat{\phi} = \frac{\hat {z} \times \hat {r} }{\sin \theta}=- \sin \phi \hat{i} + \cos \phi \hat{j}</math>
 :<math>\hat{\theta} = \hat{\phi} \times \hat {r} =-\cos \phi \cos \theta \hat{i} - \sin \phi \cos \theta  \hat{j} + \sin \theta \hat{k}</math>