91Ó°ÊÓ

A vector field is an assignment of a vector to each point in a subset of space. In this exercise, the vector field is described by the function \( \mathbf{E} = (x + 2y) \mathbf{i} + (x - 3y) \mathbf{j} \). Here, each vector at point \((x, y)\) in the plane is given a specific direction and magnitude that results from this equation. Vectors in vector fields can represent many physical phenomena, such as fluid flow, force, or electromagnetic fields. The notation \( \mathbf{i} \) and \( \mathbf{j} \) represent the unit vectors in the x and y directions, respectively. This means that each vector in the field is broken down into x and y components based on the coefficients of \( \mathbf{i} \) and \( \mathbf{j} \). Understanding vector fields is crucial for evaluating line integrals, which measure the effect of the field along a specific path.

Parametric form is a way of defining a curve or a line with respect to one or more variables called parameters. In this problem, the parametric form is used to describe the straight line path from point \( A(0,0) \) to point \( B(3, 2) \). This is done by setting \( x = 3t \) and \( y = 2t \), where \( t \) ranges from 0 to 1. As \( t \) varies, the pair \((x,y)\) traces the path of the line from A to B. This method is particularly useful for simplifying the evaluation of line integrals.With the parametric form, the differential element \( \mathrm{ds} \) along the path becomes \((3 \mathbf{i} + 2 \mathbf{j}) dt\). This transformation facilitates the computation of the integral by providing a consistent way to express changes in position along the path.

The dot product, also known as the scalar product, is an algebraic operation that takes two equal-length sequences of numbers, usually coordinate vectors, and returns a single number. It is essential in computing work done or energy transferred, where force and displacement vectors are involved.In this exercise, we compute the dot product of the vector field \( \mathbf{E} \) and the differential path \( \mathrm{ds} \) during the evaluation of the line integral. Given two vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} \), their dot product is calculated as:\[\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2.\]In the solution, this results in expressions like \( 21t - 6t \), which simplifies down to \( 15t \), allowing for the actual integration step to be carried out.

A piecewise path is a route divided into different segments, each of which can be analyzed separately. In this problem, the path from \( A \) to \( B \) is broken down into two segments: along the x-axis and then vertically along the y-axis.For the first piece, the path along the x-axis, we consider \( y = 0 \). Here, the vector field \( \mathbf{E} = x \mathbf{i} \) simplifies the integral, evaluated from \( x = 0 \) to \( x = 3 \). The differential \( \mathrm{ds} \) becomes \( \mathbf{i} \, dx \).For the second piece, at \( x = 3 \), we move vertically from \( y = 0 \) to \( y = 2 \). The vector field becomes \( \mathbf{E} = (3 + 2y) \mathbf{i} + (3 - 3y) \mathbf{j} \) with \( \mathrm{ds} = \mathbf{j} \, dy \).By calculating the integral for each segment separately and then summing these results, we obtain the total line integral for the piecewise path, illustrating the flexibility of this approach when dealing with complex paths.