Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. A paddle wheel with its axis in the direction \langle 0,1,-1\rangle would not spin when put in the vector field $$ \mathbf{F}=\langle 1,1,2\rangle \times\langle x, y, z\rangle $$ b. Stokes' Theorem relates the flux of a vector field \(\mathbf{F}\) across a surface to the values of \(\mathbf{F}\) on the boundary of the surface. c. A vector field of the form \(\mathbf{F}=\langle a+f(x), b+g(y)\) \(c+h(z)\rangle,\) where \(a, b,\) and \(c\) are constants, has zero circulation on a closed curve. d. If a vector field has zero circulation on all simple closed smooth curves \(C\) in a region \(D,\) then \(\mathbf{F}\) is conservative on \(D\)

Short answer

Question: Determine which of the following statements are true or false. a) For a paddle-wheel to not spin when put in the vector field, torque acting on it must be zero. b) Stokes' theorem relates the flux through a surface to the circulation around its boundary. c) A vector field with constant terms has zero circulation around a closed curve. d) If a vector field has zero circulation on all simple closed smooth curves C in a region D, then F is conservative on D. Answer: a) False b) False c) True d) True

Step-by-step solution

Statement a: Paddle wheel and vector field

First, we need to find the vector field \(\mathbf{F}\) by taking the cross product of the given vectors. The vector field is: $$ \mathbf{F} = \langle 1, 1, 2 \rangle \times \langle x, y, z \rangle $$ So, \(\mathbf{F} = \langle 2y - z, z - 2x, x - y \rangle\). Now, we need to calculate the torque \(\boldsymbol{\tau}\) by taking the cross product of the paddle wheel axis and the vector field: $$ \boldsymbol{\tau} = \langle 0, 1, -1 \rangle \times \langle 2y - z, z - 2x, x - y \rangle $$ \(\boldsymbol{\tau} = \langle 2x+2y, x-3z, 4x-2y \rangle\). Since the torque doesn't have all zero components, the paddle wheel would spin when put in the vector field \(\mathbf{F}\). So, the given statement is false.

Statement b: Stokes' theorem

Stokes' theorem relates the curl of a vector field over a surface to the circulation of the vector field around the boundary of the surface. Mathematically, it can be expressed as: $$ \oint_C \mathbf{F} \cdot d\mathbf{r}= \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} $$ However, the statement claims that Stokes' theorem relates flux to the boundary, which is incorrect. Flux is usually associated with the divergence theorem, not Stokes' theorem. So, the given statement is false.

Statement c: Vector field with constant terms

For this statement, we have the vector field \(\mathbf{F} = \langle a + f(x), b + g(y), c + h(z) \rangle\). We need to determine if the circulation around a closed curve is zero. Circulation can be found by calculating the line integral of the vector field over the curve: $$ \oint_C \mathbf{F} \cdot d\mathbf{r} $$ If we consider a closed curve C, we will have \(\Delta x = 0, \Delta y = 0,\) and \(\Delta z = 0\). This means that the circulation around the closed curve C is: $$ \oint_C \mathbf{F} \cdot d\mathbf{r} = a \Delta x + b \Delta y + c \Delta z = 0 $$ So, the circulation around the closed curve C is zero. Therefore, the given statement is true.

Statement d: Zero circulation and conservative fields

The statement says that if a vector field has zero circulation on all simple closed smooth curves \(C\) in a region \(D\), then \(\mathbf{F}\) is conservative on \(D\). A conservative vector field is one where the net work done around a closed loop is always zero. In mathematical terms, a vector field is conservative if and only if: $$ \oint_C \mathbf{F} \cdot d\mathbf{r} = 0 $$ Since this condition holds for all simple closed smooth curves \(C\) in the region \(D\), the given statement is true.

Key concepts

Cross Product

Understanding the cross product is crucial when working with vector fields. The cross product of two vectors results in a vector that is perpendicular to both of the original vectors. This new vector's direction is determined by the right-hand rule, and its magnitude represents the area of the parallelogram formed by the two vectors. In mathematical terms, if we have vectors \( \mathbf{A} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{B} = \langle b_1, b_2, b_3 \rangle \), their cross product \( \mathbf{A} \times \mathbf{B} \) is given by:

• \( (a_2b_3 - a_3b_2) \)
• \( (a_3b_1 - a_1b_3) \)
• \( (a_1b_2 - a_2b_1) \)

The cross product is particularly useful in physics and engineering for determining torques, rotational forces, and other vector-related calculations. In the original exercise, understanding the cross product is essential to determining whether a paddle wheel will spin when placed in a vector field. By calculating the cross product of the paddle's axis and the vector field, we can find the torque acting on the wheel.

Stokes' Theorem

Stokes' Theorem is an important result in the field of vector calculus. It connects the surface integral of the curl of a vector field over a surface to the line integral of the vector field around the boundary of the surface. In simpler terms, Stokes' Theorem allows us to relate circulation around a boundary to the behavior of a field over a region. Mathematically, it's expressed as:\[ \oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (abla \times \mathbf{F}) \cdot d\mathbf{S} \]In this formula:

• \( \oint_C \mathbf{F} \cdot d\mathbf{r} \) is the line integral over the boundary \( C \)
• \( \iint_S (abla \times \mathbf{F}) \cdot d\mathbf{S} \) is the surface integral over the surface \( S \)

Using Stokes' Theorem, you can calculate one type of integral when you have enough information to compute the other. The theorem does not relate to the flux, which is more often associated with the Divergence Theorem. The original question mistakenly associates Stokes' Theorem with flux, which is why that statement is false.

Conservative Fields

Conservative vector fields have special properties. A vector field \( \mathbf{F} \) is called conservative if there exists a scalar potential function \( \phi \) such that \( \mathbf{F} = abla \phi \). In practical terms, this means the line integral of \( \mathbf{F} \) over any closed path is zero, which corresponds to the idea that the net work done by a conservative force around any closed loop is zero. Mathematically:\[ \oint_C \mathbf{F} \cdot d\mathbf{r} = 0 \]This holds true for all closed curves \( C \) within a given region. An example of a conservative field would be the gravitational field, where gravitational forces are path-independent. Hence, any work done moving an object in a closed loop within this field is always zero. In relation to the exercise, if a vector field displays zero circulation for all simple closed curves within a region \( D \), then that vector field is indeed conservative. Understanding this can help solve a variety of complex problems in physics and engineering.

Line Integrals

Line integrals are a fundamental concept in calculus used to measure the effect of a vector field along a curve. They extend the idea of integrating with respect to length, to take into account how the field interacts with the path. The line integral of a vector field \( \mathbf{F} \) along a curve \( C \) is given by:\[ \oint_C \mathbf{F} \cdot d\mathbf{r} \]This integral represents the cumulative effect of the vector field on the curve, summing up how much the field 'pushes' along the path. Line integrals have applications in various fields such as physics, often representing work done by a force as an object moves through a field. They are also crucial in understanding the circulation of a field, particularly in verifying if a field is conservative. In the exercise, determining zero circulation with line integrals helps in identifying conservative fields.