Chapter 4: Partial Differentiation
Section 4.1 - First-Order Partial Derivatives
Section 4.2 - Higher-Order Partial Derivatives
Section 4.3 - Chain Rule
Section 4.4 - Directional Derivative
Section 4.5 - Gradient Vector
Section 4.6 - Surface Normal and Tangent Plane
Section 4.7 - Approximations
Section 4.8 - Unconstrained Optimization
Section 4.9 - Constrained Optimization
Section 4.10 - Optimization on Closed Domains
Section 4.11 - Differentiability
Chapter 4 treats differentiation of functions of several variables. Derivatives of such functions are called partial derivatives, use the special notation ∂ instead of d, and are de facto what the diff command delivers. When Maple takes an ordinary derivative, it holds constant all variables except the differentiation variable. That is precisely the notion of a partial derivative. As with ordinary derivatives, there are higher-order partial derivatives. For nearly every function met in a first course in multivariate calculus, the order in which these higher-order partial derivatives are taken does not matter.
Thinking of a function such as w=gx,y,z, where w represents a physical quantity such as temperature, the question "At what rate does the temperature vary in a specified direction?" leads to the notion of the directional derivative. The directional derivative can be expressed in terms of the gradient vector, a vector orthogonal to the level sets (either curves or surfaces, depending on the number of independent variables). The orthogonality of the gradient vector supports calculations of tangent planes for surfaces and tangent lines for curves.
Of course, differentiation is at the heart of techniques for finding extrema for functions of several variables. As in single-variable calculus, optimization problems can be either unconstrained, or constrained. For unconstrained optimization problems, Maple has the SecondDerivativeTest command for classifying the nature of any critical points found by differentiation techniques. Constrained optimization problems yield to the Lagrange multiplier technique, which Maple implements with its LagrangeMultipliers command.
Finally, the last section clarifies a fundamental difference between the differential calculus of one and several variables. In single-variable calculus, the derivative is defined first, and the differential is given in terms of the derivative. In multivariate calculus, the differential is defined first, and functions that have a differential are said to be differentiable. In either case, it remains true that functions that are differentiable are continuous.
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