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In the realm of calculus, partial derivatives play a crucial role in understanding how multivariable functions change. They measure how a function changes as one particular variable changes, holding others constant.

For a vector function like \(\mathbf{f}(x, y) = \left[\begin{array}{c} (x-y)^{2} \ 2 x^{2} y \end{array}\right]\), we focus on finding the derivative of each individual element with respect to one variable at a time. Consider the first component \((x-y)^2\). The partial derivative with respect to \(x\), denoted as \(f_x\), would be \(2(x-y)\), because the term involves \(x\) linearly. The partial derivative with respect to \(y\), \(f_y\), is \(-2(x-y)\), due to the influence of \(y\) being subtracted.

For the second component, \(2x^2y\), the partial with respect to \(x\) results in \(4xy\) because it considers the quadratic term in \(x\). Meanwhile, \(g_y = 2x^2\) reflects the linearity in \(y\). Through these derivatives, we gauge how each part of the function varies independently with changes in \(x\) or \(y\), forming the basis for linear approximation.

Vector functions extend the idea of regular functions to multiple dimensions. Instead of mapping an input to a single output value, vector functions map inputs to vectors, providing outputs in multiple dimensions simultaneously.

Consider our vector function example, \(\mathbf{f}(x, y) = \left[\begin{array}{c} (x-y)^{2} \ 2 x^{2} y \end{array}\right]\). Here, for every input pair \((x, y)\), we get a vector with two entries. Each entry represents different functional outputs based on the given inputs.

This type of function is invaluable in fields requiring calculations across multiple dimensions, such as physics for force vectors, or computer graphics for transformations. Understanding how to work with and analyze vector functions, especially in the context of partial derivatives and approximations, enables us to solve complex real-world problems by breaking them down into multiple, comprehensible parts.

Function evaluation involves finding the function's value at a particular point, using a set of input variables. It can be considered the method of verifying and interpreting mathematical expressions in practical terms.

For the function \(\mathbf{f}(x, y) = \left[\begin{array}{c} (x-y)^{2} \ 2 x^{2} y \end{array}\right]\), evaluating at a point like \((2, -3)\) involves substituting \(x\) with 2 and \(y\) with -3.

• The first component: \((2 - (-3))^2 = 25\)
• The second component: \(2 \cdot 2^2 \cdot (-3) = -24\)

Thus, the evaluated function at \((2, -3)\) gives the vector \(\begin{bmatrix} 25 \ -24 \end{bmatrix}\). Accurately evaluating functions is essential to understanding their behavior at specific points and serves as the basis for further analytical methods like differentiation and integration.

Non-linear functions are those where the relationship between the input and output variables is not a straight line. They exhibit a wide range of behaviors and include terms such as squares, cubes, and other polynomial expressions.

In our context, \(\mathbf{f}(x, y) = \left[\begin{array}{c} (x-y)^{2} \ 2 x^{2} y \end{array}\right]\) is a clear example of a non-linear function. The presence of exponentiation in terms like \((x-y)^2\) and \(x^2\) confirms the non-linearity.

Non-linear functions often require more sophisticated tools for analysis, such as differentiation and linear approximation. Approximating a non-linear function with a linear one near a point simplifies understanding complex behaviors locally. This is the essence of linear approximation, which acts as a powerful tool in calculus to handle non-linear functions efficiently, especially in fields like data modeling, physics, and engineering.