Question

Compute the value of \(\int_{A}^{P} \mathbf{G} \cdot d \mathbf{L}\) for \(\mathbf{G}=2 y \mathbf{a}_{x}\) with \(A(1,-1,2)\) and \(P(2,1,2)\) using the path ( \(a\) ) straight-line segments \(A(1,-1,2)\) to \(B(1,1,2)\) to \(P(2,1,2) ;(b)\) straight-line segments \(A(1,-1,2)\) to \(C(2,-1,2)\) to \(P(2,1,2)\)

Short answer

#tag_title#Step 2: Compute the line integral #tag_content#First, let's find the derivatives of the parametric paths: \(d\mathbf{r}_1 = \begin{bmatrix} 0 \\ dt \\ 0 \end{bmatrix}\), \(d\mathbf{r}_2 = \begin{bmatrix} dt \\ 0 \\ 0 \end{bmatrix}\), \(d\mathbf{r}_3 = \begin{bmatrix} dt \\ 0 \\ 0 \end{bmatrix}\), and \(d\mathbf{r}_4 = \begin{bmatrix} 0 \\ dt \\ 0 \end{bmatrix}\) Now, let's compute the line integral for both paths: For path a: \(\int_{A}^{P} \mathbf{G} \cdot d \mathbf{L} = \int_{A}^{B} \mathbf{G} \cdot d \mathbf{L} + \int_{B}^{P} \mathbf{G} \cdot d \mathbf{L}\) \(= \int_{0}^{2}(2(-1+t)\mathbf{a}_x) \cdot \left(\begin{bmatrix} 0 \\ dt \\ 0 \end{bmatrix}\right) + \int_{0}^{1}(2\mathbf{a}_x) \cdot \left(\begin{bmatrix} dt \\ 0 \\ 0 \end{bmatrix}\right)\) \(= 2\int_{0}^{2} (-1+t)dt + 2\int_{0}^{1} dt\) \(= 2[-t + \frac{1}{2}t^2]_0^2 + 2[t]_0^1\) \(= 2[-4 + 4] + 2[1] = 0 + 2 = 2\) For path b: \(\int_{A}^{P} \mathbf{G} \cdot d \mathbf{L} = \int_{A}^{C} \mathbf{G} \cdot d \mathbf{L} + \int_{C}^{P} \mathbf{G} \cdot d \mathbf{L}\) \(= \int_{0}^{1}(2y\mathbf{a}_x) \cdot \left(\begin{bmatrix} dt \\ 0 \\ 0 \end{bmatrix}\right) + \int_{0}^{2}(4\mathbf{a}_x) \cdot \left(\begin{bmatrix} 0 \\ dt \\ 0 \end{bmatrix}\right)\) \(= 2\int_{0}^{1} (-1)dt + 4\int_{0}^{2} dt\) \(= 2[-t]_0^1 + 4[t]_0^2\) \(= 2[-1] + 4[2] = -2 + 8 = 6\) So, for path a, the line integral \(\int_{A}^{P} \mathbf{G} \cdot d \mathbf{L} = 2\) and for path b, the line integral \(\int_{A}^{P} \mathbf{G} \cdot d \mathbf{L} = 6\).

Step-by-step solution

Parametrize the paths

For Path a, we have 2 segments: \(A(1,-1,2)\) to \(B(1,1,2)\), and \(B(1,1,2)\) to \(P(2,1,2)\) Segment 1: \(\mathbf{r}_1(t) = \begin{bmatrix} 1 \\ -1+t \\ 2 \end{bmatrix}\) for \(t \in [0, 2]\) Segment 2: \(\mathbf{r}_2(t) = \begin{bmatrix} 1+t \\ 1 \\ 2 \end{bmatrix}\) for \(t \in [0, 1]\) For Path b, we have 2 segments: \(A(1,-1,2)\) to \(C(2,-1,2)\), and \(C(2,-1,2)\) to \(P(2,1,2)\) Segment 1: \(\mathbf{r}_3(t) = \begin{bmatrix} 1+t \\ -1 \\ 2 \end{bmatrix}\) for \(t \in [0, 1]\) Segment 2: \(\mathbf{r}_4(t) = \begin{bmatrix} 2 \\ -1+t \\ 2 \end{bmatrix}\) for \(t \in [0, 2]\)

Key concepts

Vector Fields

In mathematics, a vector field is a representation where each point in a plane or space is associated with a vector. This means it has both a magnitude and a direction. For example, in our exercise, the vector field is defined by \( \mathbf{G}=2y \mathbf{a}_{x} \). Here, \( \mathbf{G} \) assigns a vector based on the position \( y \) in the coordinate plane.

Vector fields are extremely useful for visualizing the flow of quantities, such as fluid dynamics and electromagnetics. They're crucial in understanding how things like wind or electric fields behave over a region.

• Magnitude: Describes the size of the vector.
• Direction: Indicates where the vector is pointing.
• The function itself can often vary with position, just as \( \mathbf{G} \) varies with \( y \).

Understanding vector fields helps us better interpret the science of motion and interactions in various physical and abstract spaces.

Path Parametrization

Path parametrization simplifies the description of a movement path by using a single variable, often denoted as \( t \). In our case, the path from point \( A \) to point \( P \) is broken into segments with parametrized functions that indicate positions along the path as \( t \) changes.

The concept of parametrization allows us to represent complex movements with simpler equations. In this exercise, we detailed path segments using specific formulas like \( \mathbf{r}_1(t) = \begin{bmatrix} 1 \ -1+t \ 2 \end{bmatrix} \) for the first segment of path (a), where \( t \) moves from 0 to 2.

• Simplifies the analysis of a path.
• Makes calculations more manageable.
• Allows a systematic approach to solve integrals over paths.

Consider path parametrization as a way to map the journey detail into manageable equations that describe every twist and turn smoothly.

Electromagnetics

Electromagnetics studies electric and magnetic fields and their interactions with charged particles. Vector fields, such as the one in our exercise, are pivotal in this discipline. A fundamental operation in electromagnetics is evaluating line integrals, similar to those seen in our problem, which calculate the work done by or against fields like \( \mathbf{G}=2y \mathbf{a}_{x} \).

Line integrals in electromagnetics can model phenomena like:

• The work done by electric fields on charged particles.
• Induced voltages around loops in varying magnetic fields.
• The electromagnetic force experienced by particles moving through space.

This interrelationship emphasizes how the mathematical framework of line integrals directly supports and delineates physical principles, allowing us to compute and predict electromagnetic behaviors effectively.

Integral Calculus

Integral calculus is a critical branch of calculus focused on accumulation and the area under curves. A line integral extends integral calculus into evaluating integrals along curves or paths in vector fields, rather than just over straight lines or simple areas.

In the context of this exercise, the integral \( \int_{A}^{P} \mathbf{G} \cdot d \mathbf{L} \) measures how a vector field interacts with a path. The dot product \( \mathbf{G} \cdot d \mathbf{L} \) represents the component of the field along the path direction, and integrating over the path computes the total accumulation of this effect.

• Different from traditional integrals, as it considers vector fields.
• Helps quantify physical quantities across paths, like energy or work.
• Combines both the path structure and field dynamics for comprehensive analysis.

Mastery of line integrals in integral calculus is crucial for solving complex problems in physics and engineering, providing powerful tools to evaluate real-world phenomena.