The video uses h∗v because the directional derivative is defined as an infinitesimal rate of change. Using the full vector v would represent a finite step, whereas scaling it by a small scalar h and taking the limit as h approaches 0 captures the instantaneous slope in that direction.
Conditions: The function is differentiable at the point.; The direction is specified by a vector v.; h is a scalar approaching 0.
The video uses h∗v because the directional derivative is defined as an infinitesimal rate of change. Using the full vector v would represent a finite step, whereas scaling it by a small scalar h and taking the limit as h approaches 0 captures the instantaneous slope in that direction.
Conditions: The function is differentiable at the point.; The direction is specified by a vector v.; h is a scalar approaching 0.
Yes, the dot-product formula w·∇f generalizes naturally to higher dimensions. If the input has five variables, both the direction vector and the gradient vector simply expand to have five components, maintaining the same structural relationship.
Conditions: The function is differentiable in higher dimensions.; The direction vector and gradient are defined in the same dimensional space.
Yes, the dot-product formula w·∇f generalizes naturally to higher dimensions. If the input has five variables, both the direction vector and the gradient vector simply expand to have five components, maintaining the same structural relationship.
Conditions: The function is differentiable in higher dimensions.; The direction vector and gradient are defined in the same dimensional space.