91Ó°ÊÓ

A line integral, also know as a path integral, is a type of integral where we integrate over a curve. It extends the concept of a regular integral to functions defined along a path. Unlike regular integrals that deal with functions of one variable, line integrals can work on functions over curves in multi-dimensional space. In our exercise, we're dealing with a path in three dimensions, defined by the vector function \( \mathbf{r}(t) = t \mathbf{i} + t \mathbf{j} + t \mathbf{k} \).
This path is essentially a line along the vector \( \langle 1, 1, 1 \rangle \), starting from the point \((a, a, a)\) and ending at \((b, b, b)\). When setting up the line integral in the exercise, we integrate a scalar field \( f(x, y, z) = \frac{x+y+z}{x^2+y^2+z^2} \) along this path.

• You substitute the path coordinates \( x=t, y=t, z=t \) into the scalar field, making it a function of \( t \), allowing you to compute the integral over the interval \( (a, b) \).
• The key idea is understanding how a line integral evaluates the cumulative sum of the function's values along the path in space, capturing the contribution of each point space.

In practical application, this is particularly useful in fields like physics, where you calculate quantities such as work done by a force along a path.

Parametric equations are a powerful tool in vector calculus, allowing us to represent curves in multidimensional space. By expressing coordinates \( x, y, z \) based on a common variable \( t \), we can describe complex paths and motions simply. This becomes clear when considering the vector path \( \mathbf{r}(t) = t \mathbf{i} + t \mathbf{j} + t \mathbf{k} \) in our exercise.
In this setup, each component of the vector is expressed individually as a function of \( t \):

• \( x(t) = t \)
• \( y(t) = t \)
• \( z(t) = t \)

Using these parametric equations, we transform a geometric path into a more manageable form for integration, by having a clear mapping of how each dimension behaves with respect to \( t \).
Parametric equations simplify integration by rotating the original multi-variable integral into a single-variable integral, which makes the computation significantly more straightforward.

The natural logarithm, denoted as \( \ln(t) \), is a fundamental mathematical function with properties that are crucial in calculus, especially in integration. Its relevance is seen in our exercise where the function \( f(t) = \frac{1}{t} \) is integrated. The indefinite integral of \( \frac{1}{t} \) is \( \ln |t| + C \), where \( C \) is the constant of integration.
Key properties of the natural logarithm include:

• \( \ln(1) = 0 \)
• \( \ln(e) = 1 \), where \( e \) is the base of the natural logarithm, approximately equal to 2.718.
• The natural logarithm transforms multiplication into addition: \( \ln(a) + \ln(b) = \ln(ab) \).

In the context of definite integrals, as seen in the exercise, we use: \[ \int_{a}^{b} \frac{1}{t} \ dt = \left[ \ln |t| \right]_{a}^{b} = \ln |b| - \ln |a| \] Using properties of the logarithm, this can be simplified to \( \ln \left(\frac{|b|}{|a|}\right) \). This simplification underscores the efficiency and elegance of logarithmic manipulation in calculus.