What is the two-dimensional Laplace equation shown at the start of the lecture?

The symbol $\Delta u$ represents the Laplacian of the scalar field $u$. It is a compact notation for the sum of the unmixed second partial derivatives of $u$ with respect to all coordinate directions.

Conditions: $u$ is a scalar function of multiple variables.; The coordinates are orthogonal Cartesian coordinates (implied by the simple sum of second derivatives).; The notation generalizes the 2D case to $n$ dimensions.

To find the total differential of $z = x^2y^2$, compute the partial derivatives with respect to x and y, then substitute them into the total differential formula. The result is $dz = 2$xy^$2 dx + 2x^2y dy$.

Conditions: The function is $z = x^2y^2$.; Standard rules of partial differentiation apply.

The first column of the Jacobian matrix records the output components resulting from an infinitesimal step in the x direction. Specifically, the top entry is the horizontal change in the first output component, and the bottom entry is the vertical change in the second output component, both caused by the x-step.

Conditions: The input perturbation is taken strictly in the positive x direction.; The output is a 2D vector with components $f_1$ and $f_2$.; The step size is infinitesimal, allowing linear approximation.

The Jacobian matrix is a 2x2 array of first-order partial derivatives. Its rows correspond to the output components $f_1$ and $f_2$, and its columns correspond to the input variables x and y.

Conditions: The function is a vector-valued map from $R^2$ to $R^2$.; The output components are $f_1(x,y) = x + \sin (y)$ and $f_2(x,y) = y + \sin (x)$.; The matrix is evaluated at a specific point, such as (-2, 1), to describe local linear behavior.

The notation $\nabla^2$ (nabla squared) is mathematically equivalent to $\Delta$ (delta) when representing the Laplacian operator. Both symbols denote the sum of the unmixed second partial derivatives of a scalar field.

Conditions: The operator acts on a scalar field $u$.; The context is vector calculus or PDEs.; $\nabla$ is the del/nabla operator.; $\Delta$ is the Laplace operator.