91Ó°ÊÓ

Vector calculus is a branch of mathematics that deals with vector fields and differentiable vector functions. It extends concepts from single-variable calculus, such as the integral and derivative, to multi-dimensional vectors.
When working with vectors, we often deal with quantities like displacement, velocity, and force, which have both magnitude and direction. In vector calculus, these are typically expressed in terms of unit vectors, such as \(\mathbf{i}, \mathbf{j}, \text{and } \mathbf{k} \), representing x, y, and z-components respectively.

• **Vector Fields**: These are mathematical representations that assign a vector to every point in a given space, like how a magnetic field is visualized.
• **Gradient, Divergence, and Curl**: Key operations in vector calculus that help describe how vectors change in a field. The gradient points in the direction of the greatest rate of increase of a scalar field.

In our problem, we address a vector valued function \(\frac{1}{t} \mathbf{i}+\frac{1}{5-t} \mathbf{j}+\frac{1}{2t} \mathbf{k}\). Each component tells us about the behavior in its respective dimension. This understanding is crucial when transitioning from scalar calculus to vector calculus.

Definite integrals are used to calculate the net accumulation of a quantity and are fundamental to both scalar and vector calculus. A definite integral is represented as \(\int_{a}^{b} f(x) \, dx\) and computes the accumulation of \(f(x)\) from \(x=a\) to \(x=b\).

In vector calculus, definite integrals are often applied to vector functions, providing a cumulative sum of all instantaneous values of the vector function over an interval.

• **Boundaries**: The limits \(a\) and \(b\) define the interval of integration.
• **Evaluation**: After finding the antiderivative, the Fundamental Theorem of Calculus helps evaluate the expression at these limits.

In the given exercise, we evaluate the vector function over the interval \([1, 4]\). We treat each vector component as a separate integral. This gives us three definite integrals that we must solve to find the entire vector integral result.

The integration of vector functions involves integrating each component separately. This is a straightforward application of definite integrals to each part of a vector.
Vector integrals often arise in physical contexts, such as calculating work done by a force or fluid flow along a path.

• **Component-wise Integration**: Involves solving the integral of each component function independently, as in the given problem.
• **Summing Results**: The final vector integral result is achieved by combining results from each component integration.

For the given integral:

• \( \int_{1}^{4} \frac{1}{t} \, dt \) yields \( \ln 4 \mathbf{i} \)
• \( \int_{1}^{4} \frac{1}{5-t} \, dt \) yields \( \ln 4 \mathbf{j} \)
• \( \int_{1}^{4} \frac{1}{2t} \, dt \) yields \( \frac{1}{2} \ln 4 \mathbf{k} \)

The final integral expresses the cumulative change in each vector component from \(t=1\) to \(t=4\). The combined result: \( \ln 4 \mathbf{i} + \ln 4 \mathbf{j} + \frac{1}{2} \ln 4 \mathbf{k} \), encapsulates the entire vector path described by the original function.