Chapter 3: Problem 21
Calculate
Short Answer
Expert verified
Question: Calculate the divergence of the vector field at the specified points in Cartesian, cylindrical, and spherical coordinates.
Solution:
a) In Cartesian coordinates, the divergence at point is .
b) In cylindrical coordinates, the divergence at point is .
c) In spherical coordinates, the divergence at point is .
Step by step solution
01
Calculate the divergence of in Cartesian coordinates#a)
To calculate the divergence of in Cartesian coordinates, we need to compute the following partial derivatives: , , and . We find the following:
Now we compute the cartesian divergence of as:
02
Evaluate the cartesian divergence of at the given point#a)
We find the cartesian divergence of at the given point by substituting the coordinates of the point into the resulting equation from Step 1:
03
Calculate the divergence of in cylindrical coordinates#b)
In cylindrical coordinates, we have . To compute the divergence, we use:
Computing the partial derivatives and applying the formula, we get:
04
Evaluate the cylindrical divergence of at the given point#b)
The given point in cylindrical coordinates is . We convert the angle to radians: . Now we substitute the coordinates into the divergence equation from Step 3:
05
Calculate the divergence of in spherical coordinates#c)
In spherical coordinates, we have . To compute the divergence, we use:
Computing the partial derivatives and applying the formula, we get:
06
Evaluate the spherical divergence of at the given point#c)
The given point in spherical coordinates is . We convert the angles to radians: and . Now we substitute the coordinates into the divergence equation from Step 5:
So the divergences at the specified points are:
a)
b)
c)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cartesian Coordinate System
In the study of vector calculus, the Cartesian coordinate system is fundamental. It involves three orthogonal axes: the x-axis, y-axis, and z-axis, which allow for representation of points in a three-dimensional space using a set of three coordinates .
In this exercise, we calculate the divergence of a vector field in Cartesian coordinates, which requires the computation of partial derivatives with respect to each of these three axes. For example, finding the divergence at a point involves substituting the specific x, y, and z values into the divergence formula. This system is especially useful because it simplifies to straightforward partial derivatives and simple multiplication when applying the divergence operator.
In this exercise, we calculate the divergence of a vector field in Cartesian coordinates, which requires the computation of partial derivatives with respect to each of these three axes. For example, finding the divergence at a point involves substituting the specific x, y, and z values into the divergence formula. This system is especially useful because it simplifies to straightforward partial derivatives and simple multiplication when applying the divergence operator.
Cylindrical Coordinate System
When dealing with symmetries like those found in cylinders, the cylindrical coordinate system is more beneficial. It expresses a point in space with three coordinates , where is the radial distance from the origin, is the azimuthal angle about the z-axis, and is the same as in the Cartesian system.
In this system, to calculate divergence, we need to know how to compute the derivatives with respect to and and understand how to adjust for the coordinate system's radial nature. This is evident in the exercise where specific formulas for divergence in cylindrical coordinates are employed, which take into account these nuances to evaluate the vector field.
In this system, to calculate divergence, we need to know how to compute the derivatives with respect to
Spherical Coordinate System
For problems involving spherical symmetry, the spherical coordinate system is extremely effective. Here, a point in space is described by three coordinates , where is the radial distance from the origin, is the polar angle from the z-axis, and is the azimuthal angle in the x-y plane.
The divergence in spherical coordinates requires an understanding of this system's complexity, as seen in the provided exercise. The divergence operator in spherical coordinates includes derivatives with respect to each of these coordinates, and it requires specific multiplicative factors, such as and to account for the changing volume element sizes in spherical coordinates.
The divergence in spherical coordinates requires an understanding of this system's complexity, as seen in the provided exercise. The divergence operator in spherical coordinates includes derivatives with respect to each of these coordinates, and it requires specific multiplicative factors, such as
Scalar Field Divergence
The divergence of a scalar field is not conventionally defined because divergence, by definition, applies to vector fields. However, the concept of divergence can be thought of as a measure of the rate at which the density of a physical quantity, represented by the vector field, changes in space.
In our exercise, divergence is calculated for a vector field across different coordinate systems. This divergence helps us understand how much the vector field is 'spreading out' from a point, which would be analogous to the spreading of a scalar quantity like temperature or pressure if they were modeled as vector fields.
In our exercise, divergence is calculated for a vector field across different coordinate systems. This divergence helps us understand how much the vector field is 'spreading out' from a point, which would be analogous to the spreading of a scalar quantity like temperature or pressure if they were modeled as vector fields.
Vector Calculus
Vector calculus is a mathematical discipline that deals with the differentiation and integration of vector fields, often in two or three-dimensional Euclidean space. Key operations in vector calculus include gradient, divergence, and curl.
In the context of this exercise, the focus is on the divergence operator, denoted as , which measures the magnitude of a source or sink at a given point in a vector field. This is a crucial concept for understanding physical phenomena such as the flow of fluids or the electromagnetic field generated by charges.
In the context of this exercise, the focus is on the divergence operator, denoted as
Partial Derivatives
Partial derivatives represent the rate at which a function changes as one variable changes, whilst keeping the other variables constant. In the context of the Cartesian coordinate system, we compute partial derivatives with respect to and for a vector field as shown in the exercise steps.
Determining the partial derivatives is a necessary step in calculating divergence in any coordinate system because it allows us to measure how the vector field varies in each direction around a point. It’s also essential in both cylindrical and spherical coordinates, although the variables with respect to which we differentiate change according to the system used.
Determining the partial derivatives is a necessary step in calculating divergence in any coordinate system because it allows us to measure how the vector field varies in each direction around a point. It’s also essential in both cylindrical and spherical coordinates, although the variables with respect to which we differentiate change according to the system used.