Revision history [back]
 | 1 | initial version | posted 2014-07-17 16:34:19 -0600  |
I think this is the problem with using that vector: The question is asking if there is a direction where the magnitude of the rate of change is 10, and the magnitude of the rate of change is given by the magnitude of the gradient vector. If we dot the grad vector with a non-unit vector, the dot product isn't equal to the magnitude of the rate of change, so sadly non-unit vectors don't work. In general, the directional derivative needs a unit vector for this reason.
If we normalize the vector, then it no longer makes the dot product equal to 10, so unfortunately we can't do that either. I think the only way to find it methodically is the solve the system of equations.
| | 2 | No.2 Revision |
I think this is the problem with using that vector: The question is asking if there is a direction where the magnitude of the rate of change is 10, and the magnitude of the rate of change is given by the magnitude of the gradient vector. If we dot the grad vector with a non-unit vector, the dot product isn't equal to the magnitude of the rate of change, so sadly non-unit vectors don't work. In general, the directional derivative needs a unit vector for this reason.
If we normalize the vector, then it no longer makes the dot product equal to 10, so unfortunately we can't do that either. I think the only way to find it methodically is the solve the system of equations.
Edit in response to Christina: Ah, so if he had asked "find the direction where the rate of change is 10" we'd have to solve the system, but because he only asked if it ever had that value. I completely forgot that day in class.
| | 3 | No.3 Revision |
I think this is the problem with using that vector: The question is asking if there is a direction where the magnitude of the rate of change is 10, and the magnitude of the rate of change is given by the magnitude of the gradient vector. If we dot the grad vector with a non-unit vector, the dot product isn't equal to the magnitude of the rate of change, so sadly non-unit vectors don't work. In general, the directional derivative needs a unit vector for this reason.
If we normalize the vector, then it no longer makes the dot product equal to 10, so unfortunately we can't do that either. I think the only way to find it methodically is the solve the system of equations.
Edit in response to Christina: Ah, so if he we had been asked to "find the direction where the rate of change is 10" we'd have to solve the system, but because he we were only asked if it ever had that value. value, we can just see if it ever goes that high. I completely forgot that day in class.
