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Chapter 14: Problem 4

Given a function \(\varphi,\) why is the gradient of \(\varphi\) a vector field?

Short Answer

Expert verified
Answer: The gradient of the given function \(\varphi\) can be considered a vector field because it assigns a vector, \(\left(\frac{\partial\varphi}{\partial x}, \frac{\partial\varphi}{\partial y}, \frac{\partial\varphi}{\partial z}\right)\) to each point in the space \((x, y, z)\), meeting the definition of a vector field, which is a function that maps points in some space to vectors.

Step by step solution

01

Understanding Gradient

The gradient of a scalar function is a vector-valued function that gives the direction of the maximum rate of change of the function at any given point. In other words, the gradient of a function points in the direction of the steepest increase of the function. Mathematically, the gradient is represented by the symbol \(\nabla\), and for a scalar function \(\varphi(x, y, z)\), the gradient can be written as: $$\nabla \varphi = \left(\frac{\partial\varphi}{\partial x}, \frac{\partial\varphi}{\partial y}, \frac{\partial\varphi}{\partial z}\right)$$
02

Understanding Vector Fields

A vector field is a function that assigns a vector to each point in space. Technically speaking, it's a function that maps points in some space (usually \(R^2\) or \(R^3\)) to vectors. In simple terms, a vector field can be visualized as a collection of arrows, each representing a vector, assigned to each point in a given space.
03

Gradient as a Vector Field

Now that we have an understanding of both the gradient of a function and vector fields, we can see why the gradient of \(\varphi\) is a vector field. The gradient of \(\varphi\) assigns a vector \(\left(\frac{\partial\varphi}{\partial x}, \frac{\partial\varphi}{\partial y}, \frac{\partial\varphi}{\partial z}\right)\) to each point in space \((x, y, z)\). This meets the definition of a vector field, which is a function that maps points in some space to vectors. Hence, the gradient of the given function \(\varphi\) is indeed a vector field.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector Fields
Imagine a world filled with invisible arrows shooting off in different directions at every single point. This, essentially, is a picture of a vector field. A vector field is a way of assigning a vector鈥攁 quantity with both direction and magnitude鈥攖o each point in a space. If we're considering two-dimensional space, each point like (x, y) is linked to a vector. Imagine lots of tiny arrows covering the plane, all representing a vector at their origin point.
  • In three-dimensional space, each point like (x, y, z) becomes a base for even more little arrows, pointing in various directions.
  • These vectors might describe forces like wind or gravity in physics, giving a visual grasp of how these forces act over space.

So, by thinking of vector fields, we're visually connecting every location in a space with a certain directional quantity.
Scalar Functions
Unlike vector fields, scalar functions are much simpler in a certain sense. A scalar function assigns a single value (a scalar) to each point in a space. Imagine you鈥檙e looking at a weather map, where regions are colored according to temperature values. Each location on this map is assigned a specific temperature鈥攁 scalar value.
  • For example, a mountain height map might use different shades of color to signify various elevations, marking each point with a numerical altitude value.
  • Scalar functions are like a delivery service where each address gets just one package, rather than dynamic arrows indicating directions.

When working with a scalar function like \(\varphi(x, y, z)\), we encounter all kinds of applications: representing things such as potential energy in physics, temperature distributions, or even the topography of a landscape. Scalar values remain singular, but when we analyze the changes in these values over space, things get interesting.
Partial Derivatives
In the study of calculus, particularly when dealing with functions of several variables, partial derivatives come into play. A partial derivative is the derivative of a multivariable function with respect to one variable, treating all other variables as constants. This is key when dealing with any changes specific to individual directions, one at a time.
  • Consider a function \(f(x, y)\); the partial derivative \( \frac{\partial f}{\partial x} \) measures how \(f\) changes as \(x\) changes, with \(y\) held constant.
  • Similarly, \( \frac{\partial f}{\partial y} \) tells us the sensitivity of \(f\) to changes in \(y\), while \(x\) remains fixed.

Essentially, partial derivatives allow us to take a closer look at the behavior of surfaces in multi-dimensional spaces by analyzing changes along specific axes. By combining partial derivatives, such as in the formation of a gradient, we get fuller understanding of how our function behaves at and around each point.

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Most popular questions from this chapter

Let \(\mathbf{F}\) be a radial field \(\mathbf{F}=\mathbf{r} /|\mathbf{r}|^{p},\) where \(p\) is a real number and \(\mathbf{r}=\langle x, y, z\rangle .\) With \(p=3, \mathbf{F}\) is an inverse square field. a. Show that the net flux across a sphere centered at the origin is independent of the radius of the sphere only for \(p=3\) b. Explain the observation in part (a) by finding the flux of \(\mathbf{F}=\mathbf{r} /|\mathbf{r}|^{p}\) across the boundaries of a spherical box \(\left\\{(\rho, \varphi, \theta): a \leq \rho \leq b, \varphi_{1} \leq \varphi \leq \varphi_{2}, \theta_{1} \leq \theta \leq \theta_{2}\right\\}\) for various values of \(p\)

Suppose \(y=f(x)\) is a continuous and positive function on \([a, b] .\) Let \(S\) be the surface generated when the graph of \(f\) on \([a, b]\) is revolved about the \(x\) -axis. a. Show that \(S\) is described parametrically by \(\mathbf{r}(u, v)=\langle u, f(u) \cos v, f(u) \sin v\rangle,\) for \(a \leq u \leq b, 0 \leq v \leq 2 \pi\) b. Find an integral that gives the surface area of \(S\) c. Apply the result of part (b) to find the area of the surface generated with \(f(x)=x^{3},\) for \(1 \leq x \leq 2\) d. Apply the result of part (b) to find the area of the surface generated with \(f(x)=\left(25-x^{2}\right)^{1 / 2},\) for \(3 \leq x \leq 4\).

The rotation of a three-dimensional velocity field \(\mathbf{V}=\langle u, v, w\rangle\) is measured by the vorticity \(\omega=\nabla \times \mathbf{V} .\) If \(\omega=\mathbf{0}\) at all points in the domain, the flow is irrotational. a. Which of the following velocity fields is irrotational: \(\mathbf{V}=\langle 2,-3 y, 5 z\rangle\) or \(\mathbf{V}=\langle y, x-z,-y\rangle ?\) b. Recall that for a two-dimensional source-free flow \(\mathbf{V}=(u, v, 0),\) a stream function \(\psi(x, y)\) may be defined such that \(u=\psi_{y}\) and \(v=-\psi_{x} .\) For such a two-dimensional flow, let \(\zeta=\mathbf{k} \cdot \nabla \times \mathbf{V}\) be the \(\mathbf{k}\) -component of the vorticity. Show that \(\nabla^{2} \psi=\nabla \cdot \nabla \psi=-\zeta\). c. Consider the stream function \(\psi(x, y)=\sin x \sin y\) on the square region \(R=\\{(x, y): 0 \leq x \leq \pi, 0 \leq y \leq \pi\\}\). Find the velocity components \(u\) and \(v\); then sketch the velocity field. d. For the stream function in part (c), find the vorticity function \(\zeta\) as defined in part (b). Plot several level curves of the vorticity function. Where on \(R\) is it a maximum? A minimum?

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. What is the circulation on \(C\) of the vector field \(\mathbf{F}=\langle-y,-z, x\rangle\) as a function of \(\varphi ?\) For what value of \(\varphi\) is the circulation a maximum?

The heat flow vector field for conducting objects is \(\mathbf{F}=-k \nabla T,\) where \(T(x, y, z)\) is the temperature in the object and \(k>0\) is a constant that depends on the material. Compute the outward flux of \(\mathbf{F}\) across the following surfaces S for the given temperature distributions. Assume \(k=1\). \(T(x, y, z)=100 e^{-x-y} ; S\) consists of the faces of the cube \(|x| \leq 1,|y| \leq 1,|z| \leq 1\).
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