Matematika Teknik II: 222 Kalkulus Vektor - Medan Skalar dan Medan Vektor

In this session, we will discuss scalar fields and vector fields, continuing from the previous topic of vector calculus. We will define what scalar fields and vector fields are and highlight their functions. Scalar functions relate independent variables to scalar quantities, while vector functions relate independent variables to vector quantities. For instance, for two points in Cartesian coordinates, the distance is a scalar function dependent on variables x, y, and z, while the function itself represents a scalar field.

The discussion continues with a physical example of two particles and the gravitational force between them, represented through their masses and positions. Using Newton's law of gravitation, we formulate the gravitational force acting on a mass, emphasizing the dependence on the distances measured using vectors. Here, the distance and gravitational force also become functions of x, y, and z variables, illustrating that they can also represent vector fields.

The conversation then shifts to specifying the gravitational force vector mathematically with its unit vector and magnitude. The gravitational force derived from two masses shows how it is a vector field, as it changes based on the positions of the masses. Key mathematical expressions reveal the vector's dependencies on multiple variables, solidifying their categorization as vector fields in physics.

Next, we analyze derivatives in scalar and vector functions, where the concept of taking derivatives remains similar. For vector functions defined in terms of time, we discuss the process for determining derivatives and emphasize that changes in vector positions occur over time. The fundamental rule for differentiation applies similarly, whether for scalar or vector scenarios.

As we reference vector functions that depend on multiple parameters, like R(t1, t2, t3), we introduce the notion of partial derivatives. With R as a function of three variables, we present how to compute the partial derivatives respective to each variable, enhancing our understanding of how variations occur within multi-variable contexts.

An example shows how to compute both the first and second-order partial derivatives of a vector function, maintaining the convention of treating other variables as constants during differentiation. The implication is that each differentiation yields results that are functions of the remaining variables, reiterating the relationships established between the variables.