What is the limit definition of the directional derivative shown in the video?
Conditions
- The function f is defined at (, ) and nearby points.
- The direction l is specified by angle alpha.
- rho represents the displacement along the direction l in the xy-plane.
- The limit is taken as rho -> 0.
Reasoning, step by step
- Identify the change in function value: f( + rho*cos(alpha), + rho*sin(alpha)) - .
- Identify the displacement in the domain: rho.
- Form the difference quotient: [f( + rho*cos(alpha), + rho*sin(alpha)) - ] / rho.
- Take the limit of this quotient as rho approaches 0.
- Denote the result as f'_l(, ).
Example
The video displays the formula: f'_l(, ) = lim_{rho->0} [f( + rho*cos(alpha), + rho*sin(alpha)) - ] / rho.
Common misconceptions
- Thinking the denominator is the distance in 3D space rather than the planar displacement rho.
- Confusing the directional derivative with the partial derivative (which corresponds to specific directions like alpha=0 or alpha=pi/2).
- Believing the limit must be two-sided for all functions (the video implies a standard limit, but forward-only is also common in some contexts).
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