Interpreting fxy as a 'twist.'
Slide y while holding x fixed. Watch the orange tangent line tilt as its center follows the blue trace.
Surface and x-direction tangent · Drag to rotate
x-direction tangentpath: x held fixed
Look along the y-axis
Trace and tangent, viewed in the xz-plane
x-direction slope:
As y increases, does the x-direction slope increase or decrease?
Things to explore
Begin with −xy and x = 0. Sweep y upward, then compare the three reference tangent lines. Hold y fixed and move x: does the slope change?
Switch to xy. What changes about the way the tangent tilts? Try the bowl ½(x² + y²): the traces curve, but does the x-direction slope change when you move only y?
For ½x²y, compare a sweep at a negative x, at x = 0, and at a positive x. For sin(x) sin(y), explore how both the position and the value of y affect the rate of change of slope.
For the textbook example sin(x)e−y, set x = 1.75 and sweep y upward. Then compare x = 0: the sign of the mixed partial changes between these traces. The side view uses a wider vertical range for this surface.
The orange segment is tangent in the x direction. The blue curve is the path followed when x is held fixed. For ±xy, the tangent lies entirely on the surface because each coordinate trace is straight. For curved traces, the tangent generally leaves the surface.
fxy = (fx)y measures the rate of change of the x-direction slope with respect to y, with x held fixed. It is not an angular rotation rate. This interpretation of twist depends on the coordinate directions.