A function (x, y, z) is said to be harmonic in a region D in space if
Question:
A function ƒ(x, y, z) is said to be harmonic in a region D in space if it satisfies the Laplace equation
throughout D.
a. Suppose that ƒ is harmonic throughout a bounded region D enclosed by a smooth surface S and that n is the chosen unit normal vector on S. Show that the integral over S of ∇ƒ · n, the derivative of ƒ in the direction of n, is zero.
b. Show that if ƒ is harmonic on D, then
Transcribed Image Text:
V²f = V.Vf= a²fa²fa²f + + dx² dy² dz² = 0
Fantastic news! We've Found the answer you've been seeking!
Step by Step Answer:
Answer rating: 100% (5 reviews)
a To show that the integral over S of n is zero we can use the divergence theorem The divergence the...View the full answer
Answered By
Diana Muriuki
As an online math tutor, I have several years of hands-on experience working with students of all ages and skill levels. I hold a Bachelor's degree in Mathematics and a Master's degree in Education. Additionally, I have completed multiple training courses in online teaching and tutoring methods.
Throughout my career, I have worked with students in both individual and group settings, including classroom teaching, after-school tutoring, and online instruction. I am proficient in teaching a wide range of math topics, from basic arithmetic to advanced calculus and statistics.
One of my greatest strengths as a tutor is my ability to adapt my teaching style to meet the unique needs and learning styles of each individual student. I understand that every student is different, and I strive to create a comfortable and supportive learning environment that encourages growth and development.
In addition to my formal education and tutoring experience, I am also a lifelong learner with a passion for mathematics. I am constantly seeking out new resources and methods to improve my own knowledge and skills, and I believe this passion and enthusiasm helps to inspire my students as well.
Overall, my hands-on experience and proficiency as a math tutor are grounded in a combination of formal education, practical experience, and a genuine love of mathematics. I am confident in my ability to help students achieve their goals and succeed in math, and I look forward to the opportunity to work with new students and continue to grow as an educator.
0.00
0 Reviews
10+ Question Solved
Related Book For
Thomas Calculus Early Transcendentals
ISBN: 9780321884077
13th Edition
Authors: Joel R Hass, Christopher E Heil, Maurice D Weir
Question Posted:
Students also viewed these Mathematics questions
-
The random vector (X, Y) is said to be uniformly distributed over a region R in the plane if, for some constant c, its joint density is (a) Show that 1/c = area of region R. Suppose that (X, Y) is...
-
A sequence of functions fn is said to be uniformly bounded on a set E if and only if there is an M > 0 such that |fn (x) < M for all x E and all n N. Suppose that for each n N, fn: E R is...
-
Let E be a set in Rm. For each u: E R which has second-order partial derivatives on E, Laplace's equation is defined by a) Show that if u is C2 on E, then Îu = (u) on E. b) Show that if E R3...
-
What aspects of control must exist before a subsidiary is consolidated?
-
COBIT identifies 34 IT processes and a high- level approach to control over these processes. The following is the process description of PO10 Manage Projects. A program and project management...
-
Does the work performed in audit planning change in the subsequent years of auditing the same client?
-
Consider the net cash flows and salvage values shown below. Assume the alternatives can be indefinitely renewed with the same cash flows and salvage values. Using a MARR of 8%, specify the planning...
-
Martin is working to develop a preliminary costbenefit analysis for a new client-server system. He has identified a number of cost factors and values for the new system, summarized in the following...
-
Draw the relational model for this EER for vacation property rentals RENTER Renter ID First Name Middle Initial Last Name Address Phone# EMail Signs HI RENTAL AGREEMENT Agreement ID Begin Date End...
-
An aluminum tube of 3-in. outside diameter is to carry a load P of 10 kips having an eccentricity e = 0.6 in. Knowing that the stock of tubes available for use are made of aluminum alloy 2014-T6 and...
-
Use a parametrization to express the area of the surface as a double integral. Then evaluate the integral. The portion of the sphere x 2 + y 2 + z 2 = 4 between the planes z = -1 and z = 3
-
Use a parametrization to find the flux across the surface in the specified direction. Outward (normal away from the z-axis) through the portion of the cone z = x 2 + y 2 between the planes z = 1 and...
-
Try to think of - and write down - a few scenarios in which either you were a toxic handler or you might have caused toxicity in a team context. They need not be work related; they might be in a...
-
Dewey Corp. is expected to have an EBIT of $3,050,000 next year. Depreciation, the increase in net working capital, and capital spending are expected to be $240,000, $145,000, and $245,000,...
-
Bavarian Sausage is expected to pay a $1.25 dividend next year and investors expect that dividend to grow by 3% each year forever. If the required return on the stock investment is 11%, what should...
-
A fraud examiner who is investigating the disappearance of $5,000 in cash funds finds a debit in the legal expense account and a corresponding credit in the cash accounting for $5,000 but cannot find...
-
What does fresh start accounting mean? When it is utilized, how are a company's assets and its liabilities reported? Explain and support What do you know about statement of realization and...
-
In Chapter 20, we learned that to compute an approximate 90% confidence interval, the appropriate multiplier is 1.645 instead of 2.0. This works for confidence intervals for one mean or the...
-
If the amplifier indicated by the box input impedance of oo, which of the following statements are true ? has an open loop gain as well as Feedback factor (\beta = 1/ R_1\) The feedback is voltage...
-
In Problems 5770, write each expression as a single logarithm. 2 loga (5x) - log (2x + 3) 2
-
The formula can be used to find the number of years t required to multiply an investment m times when r is the per annum interest rate compounded n times a year.(a) How many years will it take to...
-
In Problems 4576, solve each exponential equation. Express irrational solutions in exact form. 25x8.5x = -16
-
Find the interval of convergence of 8 n=0 (-3)" -(x + 10)" n! (Use symbolic notation and fractions where needed. Give your answers as intervals in the form (*,*). Use the symbol co for infinity, U...
-
Subject: Chemical Engineering Calculations Instructions: 1. Find/provide any 8 chemical reactions that are solved using ICE TABLE. Note: You can just find it anywhere in internet if you want, just...
-
// Decrease the given amount from the balance. If the amount exceeds the balance, // return false since the purchase would fail. The balance should not change in that case. // Otherwise return true...
Study smarter with the SolutionInn App