Partial Derivatives Homework: Estimate Txy(6,4)

[joemama69]

Homework Statement

x & y measured in meters. Temperature is T(x,y) Temperatures are noted in table
y= 2 4 6
x
4 74 72 68
6 87 80 75
8 90 86 80

Estimate the value of T_xy(6,4)

&

T_u, where u = (i + j)/$$\sqrt{2}$$ I do no understand how to get the i and j components

Homework Equations

The Attempt at a Solution

T_x(6,4) = (86-80)/(8-6) = 3
T_x(6,6) = (80-75)/(8-6) = 2.5

T_xy = [T_x(6,4) - T_x(6,6)]/2 = [3 - 2.5]/2 = .25

 

[mighty2000]

To get the i and j components, you can use the unit vector notation to represent the direction of u. The unit vector in the i direction is [1,0] and the unit vector in the j direction is [0,1]. So, u can be expressed as u = (1/\sqrt{2})[1,1].

To get the i and j components, we can use the dot product formula: u · v = |u||v|cosθ, where u and v are vectors, |u| and |v| are the magnitudes of the vectors, and θ is the angle between the vectors. Since u is a unit vector, |u| = 1, so the formula becomes u · v = |v|cosθ.

Since u · [1,0] = |[1,0]|cos(45) = 1/\sqrt{2}, the i component of u is 1/\sqrt{2}. Similarly, the j component of u is also 1/\sqrt{2}. Therefore, u = (1/\sqrt{2})[1,1].