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Chapter 14: Problem 1

How is a vector field \(\mathbf{F}=\langle f, g, h\rangle\) used to describe the motion of air at one instant in time?

Short Answer

Expert verified
Answer: A vector field \(\mathbf{F}=\langle f, g, h\rangle\) represents the motion of air particles in 3D space at a specific instant in time by providing the velocities in each spatial direction (x, y, and z). Each component, f, g, and h, represents the velocities in the x, y, and z directions, respectively.

Step by step solution

01

Understanding a vector field

A vector field is a mathematical representation that assigns a vector to each point in a given space (in this case, in three-dimensional space). These vectors describe the properties of a quantity in that space, such as force, velocity, or flow. In the given exercise, the vector field \(\mathbf{F}=\langle f, g, h\rangle\) describes the motion of air particles in three-dimensional space.
02

Components of the vector field

In the vector field \(\mathbf{F}=\langle f, g, h\rangle\), each of the components f, g, and h represent the velocities in the x, y, and z directions, respectively. So, if we know the values of f, g, and h at a particular point (x, y, z) we can determine the direction and speed of the air particle at that location. For example, suppose we have the following vector field: \(\mathbf{F}=\langle x, y, z\rangle\). At the point (1, 1, 1), we would have the vector \(\mathbf{F}(1, 1, 1) = \langle 1, 1, 1\rangle\), which means that the air has a velocity of 1 unit in the x, y, and z directions.
03

Describing motion at one instant in time

As time progresses, the motion of air particles may change due to different factors, such as pressure, temperature, and interactions with the environment. To describe the motion of air at just one instant in time, we fix the values of f, g, and h at each point (x, y, z). This provides a "snapshot" of the air motion at that moment, allowing us to visualize and study the flow patterns. For instance, we could plot the vector field \(\mathbf{F}=\langle x, y, z\rangle\) and analyze the air flow behavior at different points in space by examining the directions and magnitudes of the vectors. In conclusion, a vector field such as \(\mathbf{F}=\langle f, g, h\rangle\) enables us to represent the motion of air particles in a 3D space at a specific moment in time by providing the velocities in each spatial direction (x, y, and z).

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Most popular questions from this chapter

Suppose a solid object in \(\mathbb{R}^{3}\) has a temperature distribution given by \(T(x, y, z) .\) The heat flow vector field in the object is \(\mathbf{F}=-k \nabla T,\) where the conductivity \(k>0\) is a property of the material. Note that the heat flow vector points in the direction opposite that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is \(\nabla \cdot \mathbf{F}=-k \nabla \cdot \nabla T=-k \nabla^{2} T\) (the Laplacian of \(T\)). Compute the heat flow vector field and its divergence for the following temperature distributions. $$T(x, y, z)=100 e^{-x^{2}+y^{2}+z^{2}}$$

Find a vector field \(\mathbf{F}\) with the given curl. In each case, is the vector field you found unique? $$\operatorname{curl} \mathbf{F}=\langle 0,1,0\rangle$$

Prove the following identities. Assume that \(\varphi\) is \(a\) differentiable scalar-valued function and \(\mathbf{F}\) and \(\mathbf{G}\) are differentiable vector fields, all defined on a region of \(\mathbb{R}^{3}\). $$\nabla \cdot(\mathbf{F} \times \mathbf{G})=\mathbf{G} \cdot(\nabla \times \mathbf{F})-\mathbf{F} \cdot(\nabla \times \mathbf{G})$$

Begin with the paraboloid \(z=x^{2}+y^{2},\) for \(0 \leq z \leq 4,\) and slice it with the plane \(y=0\) Let \(S\) be the surface that remains for \(y \geq 0\) (including the planar surface in the \(x z\) -plane) (see figure). Let \(C\) be the semicircle and line segment that bound the cap of \(S\) in the plane \(z=4\) with counterclockwise orientation. Let \(\mathbf{F}=\langle 2 z+y, 2 x+z, 2 y+x\rangle\) a. Describe the direction of the vectors normal to the surface that are consistent with the orientation of \(C\). b. Evaluate \(\iint_{S}(\nabla \times \mathbf{F}) \cdot \mathbf{n} d S\) c. Evaluate \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) and check for agreement with part (b).

One of Maxwell's equations for electromagnetic waves is \(\nabla \times \mathbf{B}=C \frac{\partial \mathbf{E}}{\partial t},\) where \(\mathbf{E}\) is the electric field, \(\mathbf{B}\) is the magnetic field, and \(C\) is a constant. a. Show that the fields $$\mathbf{E}(z, t)=A \sin (k z-\omega t) \mathbf{i} \quad \mathbf{B}(z, t)=A \sin (k z-\omega t) \mathbf{j}$$ satisfy the equation for constants \(A, k,\) and \(\omega,\) provided \(\omega=k / C\). b. Make a rough sketch showing the directions of \(\mathbf{E}\) and \(\mathbf{B}\).
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