Proving Harmonicity of g in Vector Calculus | Closed Ball & Surface Sketch

In summary, To show that g is harmonic, we can use B.W thm to express the surface integral of g over a closed ball in terms of a line integral. Then, we can calculate the length of the curve using vector calculus and substitute it into the equation to solve for f(a).
  • #1
andyb187
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Homework Statement



Let g: D-->R. D subset of R^3 be harmonic
Then for any closed ball that is a subset of D with radius >0 and its origin in D
With its surface s=(partial d)B(a,r) = {x={x1,x2,x3} st mod(x-a) = r} show that

f(a)=(1/4.pi.r^2)INTRGL: f (over s)


Homework Equations



All the standard vector calculus ones..etc Div grad curl.

The Attempt at a Solution



Just really want a sketch so I can do it my self pretty stuck on just how to go about it/layout. Think I had something going with B.W thm but doesn't look right.
 
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  • #2
[Just a sketch]1. Use B.W thm to express the surface integral of g over s in terms of a line integral. 2. Calculate the length of the curve using vector calculus. 3. Substitute into the equation above and solve.
 

Related to Proving Harmonicity of g in Vector Calculus | Closed Ball & Surface Sketch

1. What is Vector Calculus Proof?

Vector Calculus Proof is a branch of mathematics that deals with the study of vectors and their properties, such as direction, magnitude, and addition. It also involves the application of calculus concepts, such as differentiation and integration, to vectors.

2. Why is Vector Calculus Proof important?

Vector Calculus Proof is important because it provides a powerful tool for solving problems in physics, engineering, and other fields that involve vector quantities. It also helps in understanding the geometrical and physical interpretations of vector operations.

3. What are some common applications of Vector Calculus Proof?

Some common applications of Vector Calculus Proof include solving problems in mechanics, electromagnetism, fluid dynamics, and computer graphics. It is also used in optimization problems, curve fitting, and data analysis.

4. How does Vector Calculus Proof differ from regular calculus?

The main difference between Vector Calculus Proof and regular calculus is that Vector Calculus Proof involves functions of multiple variables, whereas regular calculus deals with functions of a single variable. Vector Calculus Proof also focuses on the geometric interpretation of vectors and their operations.

5. What are some key concepts in Vector Calculus Proof?

Some key concepts in Vector Calculus Proof include vector operations (such as addition, subtraction, and scalar multiplication), dot and cross products, vector functions, and line integrals. Other important topics include gradient, divergence, and curl, which are used to analyze vector fields.

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