Gradient with v_1:=1/(\sqrt(2))((1),(1)) and v_2:=1/(5)((4),(-3)) given

kaeisky9u 2022-08-13 Answered
Let g : R 2 R be totally differentiable in the point a R 2
Let v 1 := 1 2 ( 1 1 ) and v 2 := 1 5 ( 4 3 ) be normalized direction vectors in R 2
It is g v 1 ( a ) = 5 2 and g v 2 ( a ) = 1
With the info given above, how can one find out g ( a )?
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Answers (1)

Gauge Howard
Answered 2022-08-14 Author has 19 answers
Given that v 1 and v 2 are unit vectors, the the direction derivative gives
v 2 g = g v 2
and
v 2 g = g v 2
Use these to build a linear system of 2 equations in the 2 components of g, and solve. So, if
g = [ x , y ] T , then
1 2 x + 1 2 y = 5 2
and
4 5 x 3 5 y = 1
After multiplying the first equation through by 2 and second equation by 5, they becomes
x + y = 10
4 x 3 y = 5
It is trivial to solve to get x=5 , y=5
Therefore, g = [ 5 5 ]
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