Differentiability and Differentials

Calculus 3 · Axiom Academy

Where a surface is smooth, its tangent plane predicts the change in f — and is that prediction. 1. Differentiable Means Locally Flat Zoom in on a point of a smooth surface and the curve flattens onto a straight line — the cross-section of its tangent plane . A function f(x,y) is differentiable at (a,b) exactly when this happens: near the point, f is matched by the linear function whose graph is that plane, and the leftover error dies away faster than the distance to the point. the error shrinks faster than the distance 2. The Total Differential Is That Plane's Rise Step away from (a,b) by dx in the x -direction, then by dy in the y -direction, and read off how high the tangent plane climbs. Moving in x lifts you ; moving in y lifts you . Add the two rises and you get the total differential dz — the plane's predicted change in f . The rise from stepping dx in x : rate of change in x times the x -step. The rise from stepping dy in y : rate of change in y times the y -step. A concrete case: f(x,y)=x^2y at (3,2) Here f_x = 2xy and f_y = x^2 , so f_x(3,2)=12 and f_y(3,2)=9 . The total differential at that point is . Take a small step : The true change is ; the plane predicts dz . Because the surface hugs its tangent plane, the gap is tiny for small steps and shrinks faster than the step itself . Watch the true point (on the surface) and the predicted point (on the plane) close together as the step . Error analysis: a cylinder's volume

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