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The gradient of a function is a fundamental concept in multivariable calculus that is particularly useful when analyzing surfaces and their tangent planes. For a function with several variables, the gradient is a vector that points in the direction of the steepest ascent of the function at a particular point. It is composed of partial derivatives with respect to each variable.

For instance, the function from the exercise, written as \(f(x, y, z) = x^{2} + 2z^{2} - y^{2}\), has a gradient \(abla f = (f_x, f_y, f_z)\), where \(f_x, f_y, f_z\) are the partial derivatives of \(f\) with respect to \(x, y, z\), respectively. The gradient at a specific point gives the normal vector to the tangent plane at that point, which is essential in writing the equation of the tangent plane.

A normal vector is a vector that is perpendicular to a surface at a particular point. In the context of finding the equation for a tangent plane, the normal vector is vital as it dictates the orientation of the plane. It is derived from the gradient of the function, evaluated at the point of tangency, as shown in the exercise.

After evaluating the gradient at the point \( (1, 3, -2) \), the resultant normal vector is \( abla f(1, 3, -2) = (2, -6, -8) \). This normal vector is orthogonal to the tangent plane and helps define the plane's equation by giving the coefficients for the variables \(x, y, z\) in the plane equation.

Partial derivatives represent the rate at which a function changes as one of its variables changes, while all other variables remain constant. In multivariable functions, this concept is crucial for understanding how the function behaves in relation to each of its variables independently.

The partial derivatives in the exercise are denoted by \(f_x, f_y, f_z\) and calculated as \(2x, -2y, 4z\) respectively. When evaluating these at a particular point, they contribute to the components of the gradient vector, which then is used to determine the normal vector for the tangent plane equation.

Multivariable calculus extends calculus to functions of several variables. It involves concepts like partial derivatives, gradient vectors, and tangent planes, which are all intricately linked. For example, the equation of a tangent plane in three-dimensional space relies on understanding how a function changes in multiple directions, which is where multivariable calculus comes in.

The step-by-step solution provided for the exercise is a practical application of multivariable calculus, involving the use of partial derivatives to find the gradient, which leads to the normal vector that is essential for writing the tangent plane's equation. Such applications are important in fields ranging from engineering to economics, where surfaces and their tangent planes can represent many physical phenomena or optimization problems.