Question

Let S be the disk enclosed by the curve \(C: \mathbf{r}(t)=\langle\cos \varphi \cos t, \sin t, \sin \varphi \cos t\rangle,\)for \(0 \leq t \leq 2 \pi,\) where \(0 \leq \varphi \leq \pi / 2\) is a fixed angle. Consider the vector field \(\mathbf{F}=\mathbf{a} \times \mathbf{r},\) where \(\mathbf{a}=\left\langle a_{1}, a_{2}, a_{3}\right\rangle\) is a constant nonzero vector and \(\mathbf{r}=\langle x, y, z\rangle .\) Show that the circulation is a maximum when a points in the direction of the normal to \(S\).

Short answer

Solution: After calculating the cross product, tangent vector, line integral of ∮𝐹⋅𝑇, and computing the normal vector to the surface 𝑆, we can see that to maximize the circulation, we should find the value of 𝑎 that maximizes the scalar projection of 𝐹 onto 𝑛. This condition is met when the angle between the two vectors is zero, meaning 𝐹 and 𝑛 are in the same direction. Therefore, the circulation is maximum when 𝑎 points in the direction of the normal to the surface 𝑆.

Step-by-step solution

Compute the cross product F#a# x##r#

We are given \(\mathbf{a}=\left\langle a_{1}, a_{2}, a_{3}\right\rangle\) and \(\mathbf{r}(t)=\langle\cos \varphi \cos t, \sin t, \sin \varphi \cos t\rangle\). Let's compute the cross product \(\mathbf{F}=\mathbf{a} \times \mathbf{r}\): $$ \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_{1} & a_{2} & a_{3} \\ \cos \varphi \cos t & \sin t & \sin \varphi \cos t \end{vmatrix} = \left\langle a_{2} \sin \varphi \cos^{2}t-a_{3}\sin t\cos t, -a_{1}\sin\varphi\cos^2t, a_{1}\sin t \cos t - a_{2}\cos \varphi \cos^2t \right\rangle. $$

Compute the tangent vector of the curve C

We need to find the tangent vector \(\mathbf{T}(t)\) of the curve \(C\). This can be computed by taking the derivative of the position vector \(\mathbf{r}(t)\) with respect to \(t\): $$ \mathbf{T}(t) = \frac{d}{dt} \mathbf{r}(t) = \left\langle-\cos \varphi\sin t, \cos t, -\sin \varphi\sin t\right\rangle. $$

Compute the line integral of F#.#T#

Now we need to compute the circulation around the curve \(C\), which can be obtained by computing the line integral of \(\mathbf{F} \cdot \mathbf{T}\): $$ \mathrm{Circulation} = \oint_{C} \mathbf{F} \cdot \mathbf{T} dt. $$ Plug in the expressions we found for \(\mathbf{F}\) and \(\mathbf{T}\): $$ \mathrm{Circulation} = \oint_{0}^{2\pi} \left\langle a_{2} \sin \varphi \cos^{2}t-a_{3}\sin t\cos t, -a_{1}\sin\varphi\cos^2t, a_{1}\sin t \cos t - a_{2}\cos \varphi \cos^2t \right\rangle \cdot \left\langle-\cos \varphi\sin t, \cos t, -\sin \varphi\sin t\right\rangle dt. $$ Compute the dot product and integrate: $$ \mathrm{Circulation} = \oint_{0}^{2\pi} \left( -(a_{2} \sin \varphi \cos^{2}t-a_{3}\sin t\cos t)\cos \varphi\sin t+(-a_{1}\sin\varphi\cos^2t)\cos t+(a_{1}\sin t \cos t - a_{2}\cos \varphi \cos^2t)(-\sin \varphi\sin t) \right) dt. $$

Show that the circulation is maximum when a points in the direction of the normal to S

The normal vector to the surface S can be computed as \(\mathbf{n} = \frac{\partial \mathbf{r}}{\partial t} \times \frac{\partial \mathbf{r}}{\partial \varphi}\). We are given \(\frac{\partial \mathbf{r}}{\partial t} = \mathbf{T}\), so let's find \(\frac{\partial \mathbf{r}}{\partial \varphi}\): $$ \frac{\partial \mathbf{r}}{\partial \varphi} = \left\langle -\sin \varphi \cos t, 0, \cos \varphi \cos t\right\rangle. $$ Now compute the cross product: $$ \mathbf{n} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -\cos \varphi\sin t & \cos t & -\sin \varphi\sin t \\ -\sin \varphi \cos t & 0 & \cos \varphi \cos t \end{vmatrix} = \left\langle-\cos^2\varphi\cos t, (\cos^2\varphi+\sin^2\varphi)\cos^2t, -\cos^2\varphi\cos t\right\rangle. $$ To maximize the circulation, we can find the value of \(\mathbf{a}\) that maximizes the scalar projection of \(\mathbf{F}\) onto \(\mathbf{n}\) which is given by: $$ \mathrm{Projection} = \frac{\mathbf{F} \cdot \mathbf{n}}{||\mathbf{n}||}. $$ Reading the conditions for maximum circulation we notice: - The circulation will be maximum when the projection is maximum. - The projection is maximum when the angle between both vectors is zero, i.e., \(\mathbf{F}\) and \(\mathbf{n}\) are in the same direction. So, the circulation is maximum when \(\mathbf{a}\) points in the direction of the normal to \(S\).

Key concepts

Line Integrals

Line integrals are fundamental in vector calculus, particularly useful in physics and engineering to calculate work done by a force field along a path. A line integral calculates the cumulative effect of a vector field along a curve. It takes into account both the path shape and the vector field's influence at each point on the curve. In our problem, the line integral computes the circulation of the vector field, which represents the work done by the force field along the path defined by \( C \).
The process involves taking the dot product of the vector field \( \mathbf{F} \) and the tangent vector \( \mathbf{T} \) at every point along \( C \), integrating over the curve's length. The overall aim is to understand how the vector field flows along this path. Integrating these values over the entire journey provides the total circulation, a key concept when analyzing vector fields like magnetic or fluid flow.

Cross Product

The cross product is a binary operation on two vectors in three-dimensional space. The result is a vector that is perpendicular to the plane formed by the initial vectors. Mathematically represented as \( \mathbf{a} \times \mathbf{b} \), the cross product does not merely yield a numerical value but provides a vector direction that signifies significant geometric properties.
In the given problem, we calculate the cross product \( \mathbf{F} = \mathbf{a} \times \mathbf{r} \) where \( \mathbf{a} \) and \( \mathbf{r} \) symbolize the two vectors. The direction of \( \mathbf{F} \) reveals the influence of the initial vectors, highlighting how their alignment impacts the circulation. The magnitude gives insight into the area enclosed by these vectors when imagined together as sides of a parallelogram.

Normal Vector

The normal vector is crucial in understanding the orientation of a surface in space. It is perpendicular to a surface at a given point, providing critical information about the surface's geometric properties. In this solution, the normal vector \( \mathbf{n} \) to the disk \( S \) is found using the cross product of two derivatives: the tangent vector and the derivative concerning \( \varphi \).
The normal vector assists in identifying when the circulation is maximized, which occurs when the vector field aligns perfectly with this normal standpoint. Essentially, it determines the optimal alignment of the vector field to maximize the work or circulation along the given curve, satisfying the condition outlined in the problem.

Tangent Vector

Tangent vectors play a pivotal role in calculus, representing the instantaneous direction of a curve at any point in space. By differentiating the position vector \( \mathbf{r}(t) \), we derive the tangent vector \( \mathbf{T}(t) \). This provides us with a decomposition of how the path moves at each moment.
In this exercise, finding the tangent vector \( \mathbf{T}(t) \) of the curve \( C \) allows us to compute the line integral by taking the dot product of \( \mathbf{F} \) and \( \mathbf{T} \). It gives insight into how the vector field \( \mathbf{F} \) acts along the path and helps to compute the total effect as we integrate along the curve. Understanding the behavior of \( \mathbf{T}(t) \) in relation to the vector field is key not only to solving this problem but to grasping the applications of vector fields in general contexts.