Geometrical Interpretation

If P(x, y, z) be the coordinates of a point referred to rectangular axes OX, OY, OZ then the equation
                z = f(x, y)
represents a surface.

Let a plane y = b parallel to the XZ-plane pass through P cutting the surface along the curve APB give by
                Z = f(x, b).
As y remains equal to be and x varies then P moves along the curve APB and ∂z/∂x is the ordinary derivative of f(x, b), w.r.t.x

Hence ∂z/∂x at P is the tangent of the angle which the tangent at P to the section of the surface z = f(x, y) by a plane through P parallel to the plane XOZ, makes with a line parallel to the x-axis.

Similarly, ∂z/∂z at P is the tangent is the tangent of the angle which the tangent at P to the curve of intersection f the surface z = f(x,y) and the palnet x = a, makes with a line parallel to the y-axis.



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