The directional derivative of T = (1 / (2e^(r/5))) * cos(theta) along the radial distance is -1 / (10e^(r/5)) * cos(theta), evaluated at (2, π/4, 3).
The directional derivative of a scalar function T measures the rate at which the function changes with respect to a specified direction in its domain. In this case, we are interested in the directional derivative along the radial distance for the scalar function T = (1 / (2e^(r/5))) * cos(theta), where r represents the radial distance, θ is the polar angle, and ϕ is the azimuthal angle in spherical coordinates.
Mathematically, the directional derivative is computed by taking the dot product of the gradient of T and the unit vector in the direction of interest. For the radial direction, the unit vector is given by ∂/∂r, and the resulting expression is -1 / (10e^(r/5)) * cos(theta).
To evaluate this derivative at a specific point, in this case, (2, π/4, 3), we substitute the corresponding values of r, θ, and ϕ into the expression. The numerical value obtained, approximately -0.0362, signifies the rate of change of T along the radial distance at the specified point in the spherical coordinate system.