Welcome to 3D vector fields! Today we explore the function f equals x plus y plus z. A vector field assigns a vector to every point in space. For our function, the gradient is constant: one, one, one - meaning all vectors point in the same direction with equal magnitude.
Level surfaces reveal the structure of our function. For f equals x plus y plus z equals c, we get parallel planes. Each plane contains all points where the sum of coordinates equals a constant. These planes are perpendicular to the gradient direction, showing how the function increases uniformly in space.
Directional derivatives show how the function changes along different paths. The formula is the dot product of the gradient with the unit direction vector. Since our gradient is one, one, one, the maximum rate of change is square root of three, occurring when we move in the gradient direction. Other directions give smaller rates proportional to their alignment with the gradient.
Flow lines show particle motion in the vector field. Since the gradient is constant, particles move in straight lines parallel to the direction one, one, one. Each particle follows the equation r of t equals r naught plus t times the gradient vector. This creates a uniform flow pattern where all streamlines are parallel.
In summary, the vector field f equals x plus y plus z showcases fundamental concepts in vector calculus. Its constant gradient creates uniform flow, parallel level surfaces, and predictable behavior. This simple function appears in heat conduction, where temperature increases linearly with position, fluid flow in uniform fields, and optimization problems. Understanding such basic vector fields provides the foundation for analyzing more complex systems in physics and engineering.