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3-dimensional affine function approximation basis called chain rule Chapter coefficients column complex numbers compute consider continuously differentiable converges coordinate functions corresponding curve definition denote determinant differentiable function differential equation dimension direction domain dx dy eigenvectors entries equal Example Exercise Figure finite formula Fourier function 31 function defined geometric given gradient graph Green's theorem Hence inequality inner product interval iterated integral Jacobian Jacobian matrix length line integral linear combination linear function linearly independent matrix multiplication n-tuple nonzero numerical multiplication one-to-one open set orthonormal parametrically perpendicular piecewise smooth Proof Prove quadratic polynomial real numbers real-valued function rectangle region respect satisfying Section Show shown in Fig Sketch solution span subset subspace Suppose surface tangent plane tangent vector Taylor expansion Theorem Let transformation unit vector variable vector field vector space vectors xx xx and x2