Question

In Exercises 41 and \(42,\) evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) for each curve. Discuss the orientation of the curve and its effect on the value of the integral. \(\mathbf{F}(x, y)=x^{2} \mathbf{i}+x y \mathbf{j}\) (a) \(\mathbf{r}_{1}(t)=2 t \mathbf{i}+(t-1) \mathbf{j}, \quad 1 \leq t \leq 3\) (b) \(\mathbf{r}_{2}(t)=2(3-t) \mathbf{i}+(2-t) \mathbf{j}, \quad 0 \leq t \leq 2\)

Short answer

The integral values for curves \(\mathbf{r}_1(t)\) and \(\mathbf{r}_2(t)\) will give the cumulative result. If these values are equal in magnitude but opposite in sign, it shows that the orientation of the curve has an effect on the value of the line integral. The detailed values will depend on the specific integral calculations.

Step-by-step solution

Breaking Down the Function and Parametric Equations

Write \(\mathbf{F}\) and \(\mathbf{r_1}\), \(\mathbf{r_2}\) in terms of their component functions. For example, \(\mathbf{F}(x, y) = x^2\mathbf{i} + xy\mathbf{j}\), and \(\mathbf{r_1}(t) = 2t\mathbf{i} + (t-1)\mathbf{j}\), \(\mathbf{r_2}(t) = 2(3-t)\mathbf{i}+(2-t)\mathbf{j}\).

Compute the Derivative of the Curves

Differentiate \(\mathbf{r_1}(t)\) and \(\mathbf{r_2}(t)\) with respect to \(t\). Use the power rule for differentiation.

Dot Product of \(\mathbf{F}\) and \(d\mathbf{r}\)

Substitute the component functions of \(\mathbf{r_1}\) and \(\mathbf{r_2}\) into \(\mathbf{F}(x, y)\) and calculate the dot product of \(\mathbf{F}\) and \(d\mathbf{r}\) which gives a scalar function.

Evaluate the Line Integral over the Given Interval

Evaluate the integral of the scalar function obtained from the dot product operation over the given interval. The definite integral will give a numerical value.

Repeat for the Second Curve

Repeat the above steps for the second parametric curve \(\mathbf{r_2}\).

Effect of Orientation

Compare the values obtained from both integrals. If the values are equal in magnitude but have different signs, this confirms that the orientation of the curve impacts the value of the line integral.

Key concepts

Parametric Equations

Parametric equations are a powerful tool for describing curves in mathematics, particularly in the context of calculus. Rather than expressing a relationship between two variables directly as a function, parametric equations use a third variable, often denoted by t, to define both x and y in terms of this parameter. This method allows for the representation of more complex curves, including those that loop or move back on themselves, which can't be described by a single function of y as a function of x or vice versa.

For example, in a parametrization such as \(\textbf{r}_1(t) = 2t\textbf{i} + (t-1)\textbf{j}\), the vector-valued function \(\textbf{r}_1(t)\) assigns to each value of t within a specified interval a point on the curve. The advantage of this representation becomes clear when calculating line integrals, as it allows a direct substitution of the curve's equations into the integral.

Dot Product

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. In the context of line integrals, the dot product is used to project one vector along another to ascertain the extent to which two vectors 'align' with each other.

When calculating the value of a line integral, the dot product plays a central role because it simplifies the process to finding the integral of a scalar function. This operation involves taking the vector field \(\textbf{F}(x, y)\) and the differential of the curve, \(d\textbf{r}\), and combining them to obtain a scalar that varies along the path of integration. Computationally, we find the components of each vector, multiply them correspondingly, and then sum those products to get the final scalar result that's integrated over the curve.

Orientation of Curves

The orientation of a curve significantly affects the value of a line integral when evaluated along a vector field. Orientation refers to the direction in which we traverse the curve. Two parametric curves that trace the same path but in opposite directions have different orientations.

In the context of our example, comparing line integrals calculated along \(\textbf{r}_1(t)\) and \(\textbf{r}_2(t)\) highlights the effect of orientation. If these two paths describe the same curve in opposite directions, we can expect the sign of the integrals to differ while the magnitudes remain the same. This characteristic is critical in physical applications, where orientation can represent direction of force, flow, or movement, assigning a sign to the work done or the fluid flow across a boundary.

Vector Fields

A vector field is a map that assigns a vector to every point in a space. For example, the vector field \(\textbf{F}(x, y) = x^2\textbf{i} + xy\textbf{j}\) assigns a vector with x and y components described by the functions to each point in the xy-plane. Vector fields can represent various physical concepts such as gravitational, electric, and flow fields.

Calculating a line integral over a vector field involves summing the effects of the field along a specific curve or path. This process essentially asks the question: What is the cumulative effect of the vector field along this particular trajectory? It's an essential concept used in multiple branches of physics and engineering to calculate work, circulation, and flux.