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The Tangent Plane and Linear Approximation

Zoom far enough into any smooth surface and it flattens into a plane, the same way a curve straightens into its tangent line. That plane is the tangent plane, and it is the best linear approximation to the surface at the point of contact: $L(x,y) = f(x_0,y_0) + f_x(x_0,y_0)(x-x_0) + f_y(x_0,y_0)(y-y_0)$. It is the only plane that matches the surface in both height and slope at the point, which is exactly what makes it the linearization used to propagate small changes, $\Delta f \approx f_x\,\Delta x + f_y\,\Delta y$. The match is not perfect away from the point: the gap between the surface and the plane is the approximation error, and it grows quadratically with distance, governed by the curvature (the second derivatives). Drag the white point across the surface and watch the plane re-tilt to stay tangent; the panel below takes a slice along the steepest direction so you can see the tangent line kiss the curve at the point and peel away as you move off.

Figure 1. The tangent plane and linear approximation. Top: the surface z = f(x,y) in oblique projection (wireframe coloured by height) with the tangent plane (gold) touching at the draggable point. Bottom: a slice along the gradient direction, the surface (blue) and its tangent line (green), which match in value and slope at the point and separate quadratically. Method: analytic gradients; the linearization L = f + grad f . (x - x0). Source: Stewart, Calculus, 8th ed., Sec. 14.4.

WHAT TO TRY

  • Drag the white point across the surface: the tangent plane re-tilts to stay flush, and the slice below shows the tangent line staying tangent to the moving curve.
  • Watch the cross-section: the green tangent line touches the blue curve and shares its slope at the point, then peels away faster the further you look, the error growing as distance squared.
  • Move the point to the bottom of the bowl or the top of the bump: the gradient vanishes, the plane goes horizontal, and the linear approximation is flat.
  • Switch to the saddle: along one direction the curve bends above the tangent line, along the other it bends below, the two curvatures of opposite sign.