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Lagrange, who in 1762 discovered the following differential equation, known as the minimal surface equation: For a function of two vari- ables, f(x, y), the graph z = f(x,y) is a minimal surface if and only if f satisfies (1 + f²) fyy-2ƒzƒyfry + (1 + f²) ƒzz = 0. Solving this DE is no mean feat (and, luckily for you, you are not being asked to do that here!) but here is an example of a minimal surface, called Scherk's surface. It is given by the equation f(x, y) = In For x, y € (-) × (-₂). Here is a picture of Scherk's surface: COS cos y Figure 2: Scherk's surface Show that the function f giving Scherk's surface satisfies the minimal surface equation.

Lagrange, who in 1762 discovered the following differential equation, known as the minimal surface equation: For a function of two vari- ables, f(x, y), the graph z = f(x,y) is a minimal surface if and only if f satisfies (1 + f²) fyy-2ƒzƒyfry + (1 + f²) ƒzz = 0. Solving this DE is no mean feat (and, luckily for you, you are not being asked to do that here!) but here is an example of a minimal surface, called Scherk's surface. It is given by the equation f(x, y) = In For x, y € (-) × (-₂). Here is a picture of Scherk's surface: COS cos y Figure 2: Scherk's surface Show that the function f giving Scherk's surface satisfies the minimal surface equation.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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please show working out the question couldnt fit into one image so please look at both as they are the same question just over two images thanks heaps

Lagrange, who in 1762 discovered the following differential equation, known as the minimal surface equation: For a function of two vari- ables, f(x, y), the graph z = f(x, y) is a minimal surface if and only if f satisfies (1 + f²) fyy - 2ƒzƒyfxy + (1 + f²) fxx = 0. Solving this DE is no mean feat (and, luckily for you, you are not being asked to do that here!) but here is an example of a minimal surface, called Scherk's surface. It is given by the equation f(x, y) = In (COS), cos y For x, y € (-) × (-). Here is a picture of Scherk's surface: Figure 2: Scherk's surface Show that the function f giving Scherk's surface satisfies the minimal surface equation.
Transcribed Image Text:Lagrange, who in 1762 discovered the following differential equation, known as the minimal surface equation: For a function of two vari- ables, f(x, y), the graph z = f(x, y) is a minimal surface if and only if f satisfies (1 + f²) fyy - 2ƒzƒyfxy + (1 + f²) fxx = 0. Solving this DE is no mean feat (and, luckily for you, you are not being asked to do that here!) but here is an example of a minimal surface, called Scherk's surface. It is given by the equation f(x, y) = In (COS), cos y For x, y € (-) × (-). Here is a picture of Scherk's surface: Figure 2: Scherk's surface Show that the function f giving Scherk's surface satisfies the minimal surface equation.
Minimal surfaces are surfaces that exhibit remarkable and beautiful symmetry properties. In particular, they have the property that at every point the "average" curvature is zero. This means the direction in which they are most highly curved and the direction in which they are least highly curved have equal and opposite amounts of curvature. Figure 1: The catenoid, with two curves showing equal and opposite curvature at a point This is illustrated in the image above, which shows two cross-sections of the catenoid at a point, in which you can see that the cross-sections are curved equally in opposite directions. The catenoid is an example of a minimal surface, which is produced by taking the graph of the function y = cosh x and rotating it around the x-axis to produce a surface. Minimal surfaces are in fact not merely abstract mathematical objects, they can be realised physically as soap films; the surfaces created by dipping wire into soapy water. An obvious question is: how can we determine whether a surface is minimal or not? This question was first considered by Joseph-Louis
Transcribed Image Text:Minimal surfaces are surfaces that exhibit remarkable and beautiful symmetry properties. In particular, they have the property that at every point the "average" curvature is zero. This means the direction in which they are most highly curved and the direction in which they are least highly curved have equal and opposite amounts of curvature. Figure 1: The catenoid, with two curves showing equal and opposite curvature at a point This is illustrated in the image above, which shows two cross-sections of the catenoid at a point, in which you can see that the cross-sections are curved equally in opposite directions. The catenoid is an example of a minimal surface, which is produced by taking the graph of the function y = cosh x and rotating it around the x-axis to produce a surface. Minimal surfaces are in fact not merely abstract mathematical objects, they can be realised physically as soap films; the surfaces created by dipping wire into soapy water. An obvious question is: how can we determine whether a surface is minimal or not? This question was first considered by Joseph-Louis
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G i v e n space f u n c t i o n minus f open parentheses x comma y close parentheses equals ln open parentheses fraction numerator cos x over denominator cos y end fraction close parentheses

We have to prove that given function satisfies minimal surface equation.

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