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Chapter 12: Problem 80

Prove that for the plane described by \(f(x, y)=A x+B y,\) where \(A\) and \(B\) are nonzero constants, the gradient is constant (independent of \((x, y)\) ). Interpret this result.

Short Answer

Expert verified
Answer: Yes, the gradient of the plane \(f(x, y) = Ax + By\) is constant. The gradient is given by \(\nabla f = \begin{bmatrix} A\\ B \end{bmatrix}\) and is independent of coordinates (x, y), indicating a constant rate of change in any direction parallel to the plane's surface.

Step by step solution

01

Find the Partial Derivatives of f(x, y)

To prove that the gradient of the given function is constant, we need to first find the partial derivatives of the function \(f(x, y) = Ax + By\) with respect to both \(x\) and \(y\). This will help us determine the gradient of the function. The partial derivative with respect to \(x\) is given by: $$\frac{\partial f}{\partial x} = \frac{\partial (Ax + By)}{\partial x}$$ The partial derivative with respect to \(y\) is given by: $$\frac{\partial f}{\partial y} = \frac{\partial (Ax + By)}{\partial y}$$
02

Calculate the Partial Derivatives

Now, we will calculate both partial derivatives: Partial derivative w.r.t \(x\): $$\frac{\partial f}{\partial x} = A$$ Partial derivative w.r.t \(y\): $$\frac{\partial f}{\partial y} = B$$
03

Determine the Gradient

The gradient of \(f(x, y)\), denoted as \(\nabla f\), is a vector formed using the partial derivatives found in step 2. The gradient can be represented as: $$\nabla f = \begin{bmatrix} \frac{\partial f}{\partial x}\\ \frac{\partial f}{\partial y} \end{bmatrix}$$ Substitute the calculated partial derivatives into the gradient: $$\nabla f = \begin{bmatrix} A\\ B \end{bmatrix}$$
04

Prove that the Gradient is Constant

Notice that both partial derivatives (and hence the gradient) are independent of the coordinates \((x, y)\), as they only involve constants \(A\) and \(B\). Therefore, the gradient is always constant and does not change with \((x, y)\).
05

Interpret the Result

The constant gradient of the given plane is an important observation. A plane with a constant gradient is flat (it has a constant slope) and indicates a linear increase or decrease in any direction. It also implies that the rate of change of the function \(f(x, y)\) is constant in any direction parallel to the plane's surface.

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Key Concepts

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

Partial Derivatives
Partial derivatives play a crucial role in the study of multivariable calculus. They measure the rate of change of a multivariable function with respect to one of the variables while keeping the other variables constant.

In the context of the plane described by the equation f(x, y) = Ax + By, the process of finding partial derivatives involves treating the function as a univariate function with respect to each variable in turn. When calculating the partial derivative with respect to x, the coefficient A represents the rate at which f changes for a small change in x, while keeping y constant. Similarly, for y, the constant B represents the rate of change of f as y changes.

The crucial takeaway here is understanding that since A and B are constants, the resulting partial derivatives are also constants, meaning the plane's gradient is the same at all points. This concept is imperative when studying the properties of surfaces in higher dimensions and analyzing optimization problems.
Vector Calculus
Vector calculus is a branch of mathematics that deals with differentiating and integrating vector fields, typically in three-dimensional Euclidean space. The gradient is a fundamental operation in vector calculus, representing the direction and rate of steepest ascent for a scalar field.

For the given function f(x, y) = Ax + By, the gradient is determined by combining the partial derivatives with respect to x and y into a vector. This gradient vector, which in our case is \(\begin{bmatrix} A \ B \end{bmatrix}\), points in the direction where the plane has the steepest slope.

A constant gradient, as shown in the textbook solution, implies a uniform field where the magnitude and direction of the maximum rate of change remain consistent across all points. The concept of a gradient is deeply ingrained in fields such as physics, engineering, and computer graphics, where it is used to model diverse phenomena, from electromagnetism to machine learning.
Linear Functions
Linear functions are the simplest type of polynomial functions, characterized by their constant rate of change; that is, they increase or decrease at a consistent rate, creating a straight line when graphed. The general form of a linear function in two dimensions is f(x, y) = Ax + By, where A and B are coefficients that determine the slope of the line or plane.

From our textbook example, since the gradient \( abla f \) comprises both A and B and is constant, we conclude that the function describes a flat plane. The linearity of such functions makes them a cornerstone in mathematics and economics, for describing proportional relationships and predicting outcomes within certain models.

Understanding linear functions and their properties, such as being invertible and having a constant rate of change as displayed by their gradient, is essential for mathematics students as linear functions serve as the foundation for the study of more complex, nonlinear functions.

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Most popular questions from this chapter

Two resistors in an electrical circuit with resistance \(R_{1}\) and \(R_{2}\) wired in parallel with a constant voltage give an effective resistance of \(R,\) where \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}\). a. Find \(\frac{\partial R}{\partial R_{1}}\) and \(\frac{\partial R}{\partial R_{2}}\) by solving for \(R\) and differentiating. b. Find \(\frac{\partial R}{\partial R_{1}}\) and \(\frac{\partial R}{\partial R_{2}}\) by differentiating implicitly. c. Describe how an increase in \(R_{1}\) with \(R_{2}\) constant affects \(R\). d. Describe how a decrease in \(R_{2}\) with \(R_{1}\) constant affects \(R\).

Potential functions arise frequently in physics and engineering. A potential function has the property that \(a\) field of interest (for example, an electric field, a gravitational field, or a velocity field is the gradient of the potential (or sometimes the negative of the gradient of the potential). (Potential functions are considered in depth in Chapter 14 .) In two dimensions, the motion of an ideal fluid (an incompressible and irrotational fluid) is governed by a velocity potential \(\varphi .\) The velocity components of the fluid, \(u\) in the \(x\) -direction and \(v\) in the \(y\) -direction, are given by \(\langle u, v\rangle=\nabla \varphi .\) Find the velocity components associated with the velocity potential \(\varphi(x, y)=\sin \pi x \sin 2 \pi y\).

Flow in a cylinder Poiseuille's Law is a fundamental law of fluid dynamics that describes the flow velocity of a viscous incompressible fluid in a cylinder (it is used to model blood flow through veins and arteries). It says that in a cylinder of radius \(R\) and length \(L,\) the velocity of the fluid \(r \leq R\) units from the center-line of the cylinder is \(V=\frac{P}{4 L \nu}\left(R^{2}-r^{2}\right),\) where \(P\) is the difference in the pressure between the ends of the cylinder and \(\nu\) is the viscosity of the fluid (see figure). Assuming that \(P\) and \(\nu\) are constant, the velocity \(V\) along the center line of the cylinder \((r=0)\) is \(V=k R^{2} / L,\) where \(k\) is a constant that we will take to be \(k=1.\) a. Estimate the change in the centerline velocity \((r=0)\) if the radius of the flow cylinder increases from \(R=3 \mathrm{cm}\) to \(R=3.05 \mathrm{cm}\) and the length increases from \(L=50 \mathrm{cm}\) to \(L=50.5 \mathrm{cm}.\) b. Estimate the percent change in the centerline velocity if the radius of the flow cylinder \(R\) decreases by \(1 \%\) and the length \(L\) increases by \(2 \%.\) c. Complete the following sentence: If the radius of the cylinder increases by \(p \%,\) then the length of the cylinder must increase by approximately __________ \(\%\) in order for the velocity to remain constant.

Find the dimensions of the rectangular box with maximum volume in the first octant with one vertex at the origin and the opposite vertex on the ellipsoid \(36 x^{2}+4 y^{2}+9 z^{2}=36\).

Let \(x, y,\) and \(z\) be non-negative numbers with \(x+y+z=200\) a. Find the values of \(x, y,\) and \(z\) that minimize \(x^{2}+y^{2}+z^{2}\) b. Find the values of \(x, y,\) and \(z\) that minimize \(\sqrt{x^{2}+y^{2}+z^{2}}\). c. Find the values of \(x, y,\) and \(z\) that maximize \(x y z\) d. Find the values of \(x, y,\) and \(z\) that maximize \(x^{2} y^{2} z^{2}\).
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