MIT OpenCourseWare · English · ExplanationConnection evidence
Reviewed English material v1: card "Total differential formula for two variables" at 1:26 — For a differentiable function z=z(x,y), the total differential is the sum of the partial derivative with respect to x times dx and the partial derivative with respect to y times dy. In this clip the instructor states this formula before applying it to the specific polynomial example.
The Math Sorcerer · English · ExplanationConnection evidence
Reviewed English material v1: card "Definition of Total Differential" at 0:15 — For a function z=f(x,y), the total differential dz is defined as the sum of the partial derivatives multiplied by the differentials of the independent variables.
701号时钟草 · English · ApplicationConnection evidence
Reviewed English material v1: card "Identifying matrix elements from the total differential." at 8:47 — After expansion, one obtains expressions for du and dv in terms of dx and dy. The video points out that this is isomorphic to the total differential of multivariable functions; thus the four coefficients are exactly ∂x∂u, ∂y∂u, ∂x∂v, and ∂y∂v.
To find the total differential of z=x2y2, compute the partial derivatives with respect to x and y, then substitute them into the total differential formula. The result is dz=2xy^2dx+2x2ydy.
The function is z=x2y2.
Standard rules of partial differentiation apply.
Example: The instructor calculates the partial derivative with respect to x, treating y as a constant, resulting in 2xy^2. Next, he calculates the partial derivative with respect to y, treating x as a constant, yielding 2x2y. Combining these parts, the final total differential is written as dz=2xy^2dx+2x2ydy.
The total differential of a function z=f(x,y) is defined as the sum of the partial derivative of f with respect to x multiplied by dx, and the partial derivative of f with respect to y multiplied by dy. The formula is dz = (∂f/∂x)dx + (∂f/∂y)dy.
z is a function of two variables, x and y.
The partial derivatives ∂f/x and ∂f/y exist.
Example: The instructor writes the definition on the whiteboard: the total differential dz is equal to the partial derivative of f with respect to x times dx, plus the partial derivative of f with respect to y times dy.
The total differential is used in solving exact differential equations and for approximating the actual change in a function's value, where dz approximates Δz.
The context involves differential equations or function approximation.
Example: He then discusses the utility of the total differential, mentioning its role in solving exact differential equations and its use in approximating the actual change in a function's value, where dz approximates Delta z.