Problem 33
Question
Show that $$\int\left(\frac{\dot{\mathbf{r}}}{r}-\frac{\mathbf{r} \dot{r}}{r^{2}}\right) d t=\frac{\mathbf{r}}{r}+\mathbf{C}$$ where \(\mathbf{C}\) is a constant vector.
Step-by-Step Solution
Verified Answer
To show that \(\int\left(\frac{\dot{\mathbf{r}}}{r}-\frac{\mathbf{r} \dot{r}}{r^{2}}\right) dt=\frac{\mathbf{r}}{r}+\mathbf{C}\), integrate each component separately:
1. \(\int\frac{\dot{\mathbf{r}}}{r} dt = \int\frac{\mathbf{v}}{r} dt = \frac{\mathbf{r}}{r} + \mathbf{C_1}\)
2. \(-\int\frac{\mathbf{r} \dot{r}}{r^{2}} dt = -\int\frac{\mathbf{v}}{r} dt = -\frac{\mathbf{r}}{r} + \mathbf{C_2}\)
Adding them together, we get:
\(\left(\frac{\mathbf{r}}{r}+\mathbf{C_1}\right) + \left(-\frac{\mathbf{r}}{r} + \mathbf{C_2}\right) = \frac{\mathbf{r}}{r} + \mathbf{C}\),
where \(\mathbf{C} = \mathbf{C_1} + \mathbf{C_2}\) is a constant vector.
1Step 1: Identify the components of the vector function
The vector function inside the integral has two components:
1. \(\frac{\dot{\mathbf{r}}}{r}\)
2. \(-\frac{\mathbf{r} \dot{r}}{r^{2}}\)
We will integrate each of these components separately with respect to time.
2Step 2: Integrate the first component
The first component is \(\frac{\dot{\mathbf{r}}}{r}\). To integrate this with respect to time, we have:
$$\int\frac{\dot{\mathbf{r}}}{r} dt$$
Since \(\dot{\mathbf{r}}\) is just the velocity vector \(\mathbf{v}\), we can perform a simple substitution:
$$\int\frac{\mathbf{v}}{r} dt$$
Integrating term by term, we have:
$$\frac{\mathbf{r}}{r} + \mathbf{C_1}$$
where \(\mathbf{C_1}\) is a vector of integration constants.
3Step 3: Integrate the second component
The second component is \(-\frac{\mathbf{r} \dot{r}}{r^{2}}\). To integrate this with respect to time, we have:
$$-\int\frac{\mathbf{r} \dot{r}}{r^{2}} dt$$
Now, we notice that \(\frac{\mathbf{r} \dot{r}}{r} = \mathbf{v}\), so we can perform another substitution:
$$-\int\frac{\mathbf{v}}{r} dt$$
Integrating term by term, we have:
$$-\frac{\mathbf{r}}{r} + \mathbf{C_2}$$
where \(\mathbf{C_2}\) is another vector of integration constants.
4Step 4: Add the results of the two integrals together
Now we add the results of the two integrals obtained in Step 2 and Step 3:
$$\left(\frac{\mathbf{r}}{r} + \mathbf{C_1}\right) + \left(-\frac{\mathbf{r}}{r} + \mathbf{C_2}\right)$$
This simplifies to:
$$\frac{\mathbf{r}}{r} - \frac{\mathbf{r}}{r} + \mathbf{C_1} + \mathbf{C_2}$$
So, the final result is:
$$\frac{\mathbf{r}}{r} + \mathbf{C}$$
where \(\mathbf{C} = \mathbf{C_1} + \mathbf{C_2}\) is a constant vector, as required.
Key Concepts
Vector IntegrationVelocity VectorIntegration ConstantsConstant Vector
Vector Integration
In vector calculus, integrating a vector function involves finding a vector whose derivative gives the original function. This is called vector integration. Unlike scalar integration, where we compute the area under a curve, vector integration deals with the components of vector functions. Each component of the vector is integrated separately, similar to how you'd integrate a multi-variable function term by term.
Consider a vector function \[\int \left( \frac{\dot{\mathbf{r}}}{r} - \frac{\mathbf{r} \dot{r}}{r^{2}} \right) dt\]where we need to integrate its components separately with respect to time. This step-by-step approach allows us to handle complex vector functions by focusing on their components, simplifying the integration process.
By breaking down the problem into smaller parts, we deal with individual terms, making it easier to identify integration rules that apply to each. Integrating involves finding functions whose derivative matches the given term, and we handle each component of the vector following basic rules of integration, sometimes using substitutions to simplify terms further.
Consider a vector function \[\int \left( \frac{\dot{\mathbf{r}}}{r} - \frac{\mathbf{r} \dot{r}}{r^{2}} \right) dt\]where we need to integrate its components separately with respect to time. This step-by-step approach allows us to handle complex vector functions by focusing on their components, simplifying the integration process.
By breaking down the problem into smaller parts, we deal with individual terms, making it easier to identify integration rules that apply to each. Integrating involves finding functions whose derivative matches the given term, and we handle each component of the vector following basic rules of integration, sometimes using substitutions to simplify terms further.
Velocity Vector
The velocity vector, typically noted as \(\dot{\mathbf{r}}\), is a crucial concept in vector calculus and physics. It represents the rate of change of position with respect to time. When calculating integrals involving the velocity vector, it is often denoted as \(\mathbf{v}\) and plays a key role in understanding motion.
In the context of the given exercise, integrating the term \(\frac{\dot{\mathbf{r}}}{r}\) highlights how the velocity vector is related to position. By substituting \(\dot{\mathbf{r}}\) with \(\mathbf{v}\), we simplify the integral to \(\int \frac{\mathbf{v}}{r} dt\).
Understanding the velocity vector helps students visualize the movement of an object through space. As this vector changes, it describes acceleration and other motion characteristics specific to the system being analyzed. Thus, being able to integrate with respect to \(\mathbf{v}\) not only furthers understanding of the objects in motion but also how those objects are changing in velocity over time.
In the context of the given exercise, integrating the term \(\frac{\dot{\mathbf{r}}}{r}\) highlights how the velocity vector is related to position. By substituting \(\dot{\mathbf{r}}\) with \(\mathbf{v}\), we simplify the integral to \(\int \frac{\mathbf{v}}{r} dt\).
Understanding the velocity vector helps students visualize the movement of an object through space. As this vector changes, it describes acceleration and other motion characteristics specific to the system being analyzed. Thus, being able to integrate with respect to \(\mathbf{v}\) not only furthers understanding of the objects in motion but also how those objects are changing in velocity over time.
Integration Constants
Whenever you integrate, remember that every integral has an accompanying constant of integration. This is because the process of finding an antiderivative leaves an uncertainty – expressed as an arbitrary constant – about the initial value of the original function.
In vector integration, these constants are often termed as integration constants, and they appear for each component of the vector being integrated. When we solve issues involving integrals, like in our problem, you might see terms like \(\mathbf{C_1}\) and \(\mathbf{C_2}\), which are vectors of these constants.
Because integration constants help represent a family of solutions rather than just one, they allow for a broader understanding and flexibility, capturing any initial conditions that need consideration in your solution. When you add or subtract integrals, like in this exercise, where \(\mathbf{C_1}\) and \(\mathbf{C_2}\) combine, these constants of integration then simplify into a singular constant vector, representing the general solution to the problem at hand.
In vector integration, these constants are often termed as integration constants, and they appear for each component of the vector being integrated. When we solve issues involving integrals, like in our problem, you might see terms like \(\mathbf{C_1}\) and \(\mathbf{C_2}\), which are vectors of these constants.
Because integration constants help represent a family of solutions rather than just one, they allow for a broader understanding and flexibility, capturing any initial conditions that need consideration in your solution. When you add or subtract integrals, like in this exercise, where \(\mathbf{C_1}\) and \(\mathbf{C_2}\) combine, these constants of integration then simplify into a singular constant vector, representing the general solution to the problem at hand.
Constant Vector
A constant vector is one that does not change as the variables in the context do, such as time or position. In many applied problems, constant vectors can represent fixed quantities like initial velocities or positions.
In the solution of our exercise, \[\int \left( \frac{\dot{\mathbf{r}}}{r} - \frac{\mathbf{r} \dot{r}}{r^{2}} \right) dt = \frac{\mathbf{r}}{r} + \mathbf{C}\] \(\mathbf{C}\) represents a constant vector. This vector arises from the integration constants that emerged during the integration process of the vector components.
In the solution of our exercise, \[\int \left( \frac{\dot{\mathbf{r}}}{r} - \frac{\mathbf{r} \dot{r}}{r^{2}} \right) dt = \frac{\mathbf{r}}{r} + \mathbf{C}\] \(\mathbf{C}\) represents a constant vector. This vector arises from the integration constants that emerged during the integration process of the vector components.
- \(\mathbf{C}\) is independent of time and stays unchanged.
- It covers the net effect of any initial conditions or constant changes in the equation setup.
Other exercises in this chapter
Problem 31
Show that (a) \(\nabla r^{n}=n r^{(n-2)} \mathbf{r}\) (b) \(\nabla f(r)=\frac{\mathbf{r}}{r} \frac{d f}{d r}\) (c) \(\nabla^{2}(\ln r)=\frac{1}{r^{2}}\)
View solution Problem 32
Show that $$\int(2 a \mathbf{r} \cdot \mathbf{i}+2 b \mathbf{r} \cdot \mathbf{r}) d t=a \mathbf{r}^{2}+b \dot{r}^{2}+\text { const }$$ where \(\mathbf{r}\) is t
View solution Problem 35
Show that the volume common to the intersecting cylinders defined by \(x^{2}+y^{2}=a^{2}\) and \(x^{2}+z^{2}=a^{2}\) is \(V=16 a^{3} / 3\).
View solution Problem 37
Find the value of the integral \(\int_{s} \mathbf{A} \cdot d \mathbf{a},\) where \(\mathbf{A}=\left(x^{2}+y^{2}+z^{2}\right)(x \mathbf{i}+y \mathbf{j}+z \mathbf
View solution