MATH 200 Lecture Notes - Lecture 6: Unit Vector, Tangent Space

• School: Drexel University
• Department: Mathematics
• Course: MATH 200
• Professor: unknown

Document Summary

If f(x,y) is a function and (cid:1873)= is a unit vector, then the directonal derivative of f in the direction of u at (cid:4666)(cid:1876),(cid:1877)(cid:4667) is denoted (cid:4666)(cid:1876),(cid:1877)(cid:4667) and is given by (cid:4666)(cid:1876),(cid:1877)(cid:4667)= [(cid:4666)(cid:1876)+(cid:1873)(cid:2869)(cid:1871),(cid:1877)+(cid:1873)(cid:2870)(cid:1871)(cid:4667)]|(cid:1871)=(cid:882) If u is a unit vector , then (cid:4666)(cid:1876),(cid:1877),(cid:1878)(cid:4667)= [(cid:4666)(cid:1876)+(cid:1873)(cid:2869)(cid:1871),(cid:1877)+ (cid:1873)(cid:2870)(cid:1871),(cid:1878)+(cid:1873)(cid:2871)(cid:1871)]|(cid:1871)=(cid:882) If f(x,y) is differentiable at (cid:4666)(cid:1876),(cid:1877)(cid:4667) and (cid:1873)= is a unit vector then (cid:4666)(cid:1876),(cid:1877)(cid:4667) exists and (cid:4666)(cid:1876),(cid:1877)(cid:4667)=(cid:3051)(cid:4666)(cid:1876),(cid:1877)(cid:4667)(cid:1873)(cid:2869)+(cid:3052)(cid:4666)(cid:1876),(cid:1877)(cid:4667) (cid:1873)(cid:2870) If f is a function of x and y, the gradient of f is defined by (cid:4666)(cid:1876),(cid:1877)(cid:4667)= Let f be a function of 2 or 3 variables and let p=(cid:4666)(cid:1876),(cid:1877)(cid:4667)and assume f is. A) if (cid:4666)(cid:1876),(cid:1877),(cid:1878)(cid:4667)=(cid:882) then all directional derivatives of f at p are zero. B) if (cid:4666)(cid:1876),(cid:1877),(cid:1878)(cid:4667) (cid:882) then among all possible directional derivatives of f at p, the derivative in the direction of (cid:4666)(cid:1876),(cid:1877),(cid:1878)(cid:4667) has the largest value and this value is ||(cid:4666)(cid:1876),(cid:1877),(cid:1878)(cid:4667)|| To maximize (cid:4666)(cid:1876),(cid:1877)(cid:4667), pick u that is pointing in the same direction as (cid:4666)(cid:1876),(cid:1877),(cid:1878)(cid:4667) differentiable at p.