Problem 127 - Calculus Volume 3

Question

For the following exercises, evaluate the integral using the Fundamental Theorem of Line Integrals. [T] Evaluate \(\int_{C} \nabla f \cdot d \mathbf{r}\), where \(f(x, y)=x y+e^{x}\) and \(C\) is a straight line from (0,0) to (2,1)

Step-by-Step Solution

Answer

1 + e^2

Step 1: Understand the Concept

We're asked to evaluate the line integral using the Fundamental Theorem of Line Integrals. This theorem states that if a vector field is the gradient of a scalar function (as given here), then the line integral of this vector field over a curve from point \(A\) to point \(B\) equals the difference in the values of the scalar function at \(B\) and \(A\).

Step 2: Identify the Endpoints

The problem states that the curve \(C\) is a straight line from \((0,0)\) to \((2,1)\). So, point \(A = (0,0)\) and point \(B = (2,1)\).

Step 3: Evaluate the Gradient

The given scalar function is \(f(x, y) = xy + e^x\). The gradient of this function, \(abla f\), is \(\left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right) = (y + e^x, x)\). However, since we are using the Fundamental Theorem of Line Integrals, we don't need to explicitly evaluate the gradient.

Step 4: Apply the Fundamental Theorem of Line Integrals

Using the Fundamental Theorem of Line Integrals, the integral \(\int_C abla f \cdot d\mathbf{r}\) is just \(f(B) - f(A)\).

Step 5: Calculate \(f(B)\) and \(f(A)\)

Substitute \((x,y) = (2,1)\) into the function: \(f(2,1) = 2 \times 1 + e^2 = 2 + e^2\). Substitute \((x,y) = (0,0)\) into the function: \(f(0,0) = 0 \times 0 + e^0 = 1\).

Step 6: Compute the Integral

The value of the integral is \(f(B) - f(A) = (2 + e^2) - 1 = 1 + e^2\).

Key Concepts

Gradient

The concept of the gradient is fundamental when exploring line integrals. The gradient of a function is a vector that points in the direction of the greatest rate of increase of the function. It is computed as a vector of partial derivatives. If we have a scalar function like \( f(x, y) = xy + e^x \), its gradient, denoted as \( abla f \), is calculated as:

• \( \frac{\partial f}{\partial x} = y + e^x \)
• \( \frac{\partial f}{\partial y} = x \)

Thus, the gradient vector is \( (y + e^x, x) \). This vector is crucial because in the context of line integrals, it allows us to determine the directional properties of the field defined by our scalar function. The Fundamental Theorem of Line Integrals utilizes this gradient to simplify the computation process by converting it into the evaluation of the function at the boundaries of the path.

Line integrals

Line integrals are essential tools in vector calculus, particularly when you deal with vector fields. They measure the effect of a vector field along a curve or path. Consider a path \( C \) from point \( A \) to point \( B \). When you perform a line integral of a vector field over a curve, you're essentially summing the field's effects on infinitesimal pieces of the path.
The line integral of a gradient field, which is what the Fundamental Theorem of Line Integrals deals with, simplifies to the difference between the values of the original scalar function at the endpoints of the curve. This makes it significantly easier to evaluate than performing the integration directly over the path.

Scalar function

A scalar function assigns a single value to each point in space, like \( f(x, y) = xy + e^x \). Unlike vector functions that may have multiple components, a scalar function provides one output number for every input (x, y) pair you plug into it.

• For \( f(x, y) \), calculate the individual outputs directly from inputs.
• In our exercise, you find \( f(2, 1) = 2 + e^2 \) and \( f(0, 0) = 1 \).

The beauty of a scalar function in this context is that it allows us to use its inherent properties for calculations efficiently. With the Fundamental Theorem of Line Integrals, understanding the scalar function is critical as it is directly involved in providing the values needed at the endpoints for evaluating the integral.

Vector field

Vector fields are an arrangement of vectors across a plane or space indicating the directional forces acting at each point. If a vector field is defined by a gradient of a scalar function, it is called 'conservative', meaning it can be described by a potential function.
When analyzing the vector field \( abla f \) of our function \( f(x, y) = xy + e^x \), we get a vector field \( (y + e^x, x) \). The vector space now has directions and magnitudes derived from the gradient. Using this vector field with the Fundamental Theorem of Line Integrals allows simplification: instead of integrating directly over this field, calculate the scalar function only at the initial and final points of the path.