Lec 14: Non-independent variables | MIT 18.02 Multivariable Calculus, Fall 2007

The script begins with an introduction to the topic of non-independent variables, emphasizing the importance of understanding the relationships between variables in mathematical functions. It mentions the use of Lagrange multipliers to find extrema and introduces the concept of partial derivatives in the context of related variables, such as pressure, volume, and temperature in physics. The script also humorously addresses a student named Jason, instructing him to claim his package after the lecture.

This paragraph delves into the specifics of partial derivatives when variables are related by an equation, using the example of a function of three variables x, y, z constrained by g(x, y, z) = constant. The script explains how to find rates of change for one variable with respect to another while keeping others constant, even when an explicit solution for one variable in terms of the others is not available. An example using the relation x^2 yz z^3 = 8 near the point (2, 3, 1) is provided to illustrate this concept.

The script explains how to use differentials to find the rates of change for constrained variables without explicitly solving for one variable in terms of the others. It demonstrates this with the example from the previous paragraph, showing how to compute the differential of the constraint equation and set it to zero to find the relationship between the changes in x, y, and z. The paragraph also clarifies how to express dz in terms of dx and dy, and vice versa, to understand the dependencies between variables.

This section addresses potential confusion with the notation used for partial derivatives, especially in the context of change of variables. The script uses a simple example of a function f(x, y) = xy and a change of variables x = u, y = uv to illustrate how partial derivatives with respect to different variables can yield different results, highlighting the importance of specifying what is being held constant during the differentiation process.

The script introduces a new notation to explicitly indicate what is being held constant when taking partial derivatives. This is particularly important when dealing with related variables, as it avoids the contradiction that can arise from changing variables without considering the constraints. The new notation uses subscripts to denote the variables that are kept constant, providing a clearer understanding of the differentiation process.

The script presents a geometric example involving the area of a triangle with sides a and b and an angle theta. It introduces a constraint that the triangle is a right triangle, with b as the hypotenuse and relates a, b, and theta through the equation b = a * tan(theta). The paragraph explores different interpretations of how the area of the triangle depends on theta, considering various scenarios where either a or b is held constant while the other varies.

This paragraph focuses on computing the rate of change of the area of the right triangle with respect to theta, considering the constraints of the problem. It discusses three different scenarios: changing theta while keeping a and b fixed, changing theta while keeping a constant and allowing b to vary, and changing theta while keeping b constant and allowing a to vary. The script outlines the process of using differentials to find these rates of change, emphasizing the importance of considering the constraint in the calculations.

The script provides a detailed explanation of how to use differentials and the chain rule to find the rate of change of the area of the triangle with respect to theta, given the constraint that it is a right triangle. It shows how to differentiate the constraint equation to find the relationship between the differentials of a, b, and theta, and then how to substitute this relationship into the expression for the differential of the area to find the desired partial derivatives.

The final paragraph wraps up the discussion and mentions a review session for an upcoming test. It summarizes the methods covered for dealing with partial derivatives of functions with related variables and hints at further exploration of these concepts during the review session. The script encourages students to be familiar with both methods discussed, as they may be applicable to different problems or test questions.