Question

Let a path (5.1) be given and let a change of parameter be made by an equation \(t=\) \(g(\tau), \alpha \leq \tau \leq \beta\), where \(g^{\prime}(\tau)\) is continuous and positive in the interval and \(g(\alpha)=\) \(h, g(\beta)=k \cdot\) As in \((5.4)\) the line integral \(\int f(x, y) d x\) on the path \(x=\phi(g(\tau)), y=\psi(g(\tau))\) is given by $$ \int_{\alpha}^{\beta} f[\phi(g(\tau)), \psi(g(\tau))] \frac{d}{d \tau} \phi(g(\tau)) d \tau . $$ Show that this equals the integral in (5.4), so that such a change of parameter does not affect the value of the line integral.

Short answer

Question: Show that the line integral with a change of parameter is equal to the given line integral in (5.4). Solution: We verified that the line integral with the change of parameter: $$ \int_{\alpha}^{\beta} f[\phi(g(\tau)), \psi(g(\tau))] \phi'(g(\tau)) g'(\tau) d\tau $$ is equal to the given line integral in (5.4): $$ \int_{h}^{k} f[\phi(t), \psi(t)] \phi'(t) dt $$ by applying the change of parameter, using the substitution formula and comparing the two integrals. Therefore, the change of parameter does not affect the value of the line integral.

Step-by-step solution

Recall the given line integral in (5.4)

Assuming the line integral in (5.4) is of the form: $$ \int_{h}^{k} f[\phi(t), \psi(t)] \phi'(t) dt $$ We want to show that the integral in the given equation with parameter change is equal to this integral.

Apply the change of parameter

Since we have a parameter change from \(t\) to \(\tau\) with \(t = g(\tau)\), we can re-write the given integral as $$ \int_{\alpha}^{\beta} f[\phi(g(\tau)), \psi(g(\tau))] \phi'(g(\tau)) g'(\tau) d\tau. $$ We obtained this integral by substituting \(t = g(\tau)\) and using the chain rule to calculate the derivative of \(\phi\) with respect to \(\tau\).

Apply the substitution formula

To show that the new integral equals the one in (5.4), we will use the substitution formula for integrals. Let \(u = g(\tau)\), so we have \(du = g'(\tau) d\tau\). We also need to change the limits of integration according to the substitution. When \(\tau = \alpha\), we know that \(g(\alpha) = h\) and when \(\tau = \beta\), we have \(g(\beta) = k\). So, our integral becomes: $$ \int_{h}^{k} f[\phi(u), \psi(u)] \phi'(u) du $$

Compare the two integrals

Comparing our new integral with the one given in (5.4), we can see that they are identical in form: $$ \int_{h}^{k} f[\phi(g(\tau)), \psi(g(\tau))] \phi'(g(\tau)) g'(\tau) d\tau = \int_{h}^{k} f[\phi(t), \psi(t)] \phi'(t) dt $$ So, such a change of parameter does not affect the value of the line integral.

Key concepts

Change of Parameter

When performing line integrals, we often change the parameter to simplify the calculation or to suit boundary conditions. In the context of the given problem, the change of parameter involves replacing the variable \( t \) with a new variable \( \tau \), such that \( t = g(\tau) \). This change is important because it carries through the entire calculation and affects the limits of integration. The function \( g(\tau) \) must be continuous and have a continuous and positive derivative \( g'(\tau) \) in the interval

• This ensures the transformation is smooth and maintains the orientation of the path.
• The continuity and positivity of \( g'(\tau) \) are crucial for maintaining consistency throughout the integral.

Understanding this change of parameter provides flexibility in solving integrals, particularly when dealing with more complex paths or transitioning between different coordinate systems.

Substitution Formula

The substitution formula for integrals is a powerful tool when dealing with transformations, much like the change of parameter described earlier. It allows us to change variables within an integral, adjusting the limits of integration accordingly.
To apply it, the original variable \( t \) is expressed in terms of \( \tau \) through \( t = g(\tau) \). In our case, this alteration leads to the need to incorporate the derivative \( g'(\tau) \) into the differential, recognized as \( dt = g'(\tau) d\tau \).

• Consequently, the integral involving \( t \) transitions into one in terms of \( \tau \), hence \( \int f(\phi(g(\tau)), \psi(g(\tau))) g'(\tau) d\tau \).
• This substitution also necessitates a shift in the integration limits from \( [h, k] \) to \( [\alpha, \beta] \), adhering to the change in parameter.

This method ensures that the integral's value remains unchanged as long as the substitutions and limits are correctly managed.

Chain Rule

The chain rule is a fundamental concept in calculus used to differentiate composite functions, which appear frequently in parameter changes for line integrals.
In this problem, the dependent variable \( \phi(t) \) becomes \( \phi(g(\tau)) \), where \( t = g(\tau) \). To find the derivative with respect to \( \tau \), the chain rule is utilized.

• We compute the derivative \( \frac{d}{d\tau}\phi(g(\tau)) = \phi'(g(\tau))g'(\tau) \).
• This operation effectively intertwines the derivatives of \( \phi \) and \( g \), capturing the entire rate of change as respects to \( \tau \).

Using the chain rule here is crucial for accurately transforming the line integral and ensuring all necessary derivatives are accounted for. This is essential for preserving the equivalence to the original integral over \( t \).

Derivatives

Derivatives play a pivotal role when dealing with line integrals, especially after a change of parameter. They provide a measure of how a function changes with respect to its input variables. After transforming the function with a new parameter \( \tau \), derivatives allow us to understand and compute the effect of this transformation by expressing how every component of the path changes.

• In the given problem, \( f(\phi(g(\tau)), \psi(g(\tau))) \) is an example where derivatives like \( \phi'(g(\tau)) \) and \( g'(\tau) \) reveal how the function behaves in terms of \( \tau \).
• These derivatives are combined using the chain rule to express the original line integral in terms of the new parameter.

Understanding derivatives within the context of transformed parameters ensures that the integrity of the line integral is maintained and that calculations are correctly transitioned between different representations.