How do you evaluate the vector calculus question over an ellipsoid?

In summary, the question is about evaluating the integral of the dot product of the gradient of 1/r over an ellipsoid, and the conversation discusses two different answers - 0 and -4pi. The confusion arises from the fact that 1/r is not harmonic at the origin.
  • #1
BrandonATC
2
0
The question is

Let r=r(x,y,z) where it is the distance from a point O. Evaluate

[tex]\oint[/tex][tex]\nabla[/tex](1/r)*ndS

(where * is the dot product)

over the ellipsoid

x^2/4 +y^2/9+z^2/25=1

I thought the answer was 0 since the ellipsoid is a simply connected region in R^3 and the div(grad(1/r))= 0 since 1/r is harmonic.

But some people in the class said the answer was -4pi

Thanks
 
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  • #2
Welcome to PF!

Hi BrandonATC! Welcome to PF! :smile:

(have a pi: π and a del: ∇ and try using the X2 icon just above the Reply box :wink:)
BrandonATC said:
… I thought the answer was 0 since the ellipsoid is a simply connected region in R^3 and the div(grad(1/r))= 0 since 1/r is harmonic.

No, it's not harmonic at the origin. :wink:
 
  • #3
thank you, just wondering how do you show the answer is -4pi?
 

What is vector calculus?

Vector calculus is a branch of mathematics that deals with the study of vector fields and scalar fields, and their derivatives and integrals. It is widely used in many fields of science, such as physics, engineering, and computer graphics.

What are some common applications of vector calculus?

Vector calculus has many real-world applications, including determining the velocity and acceleration of moving objects, modeling fluid flow, and calculating electric and magnetic fields in electromagnetism.

What are the basic operations in vector calculus?

The basic operations in vector calculus include vector addition and subtraction, scalar multiplication, dot product, cross product, and differentiation and integration of vector fields.

How is vector calculus different from regular calculus?

While regular calculus deals with functions of one variable, vector calculus deals with functions of multiple variables. This allows for a deeper understanding of physical phenomena that involve multiple quantities and their relationships.

What are some useful tools for solving vector calculus problems?

Some useful tools for solving vector calculus problems include vector notation, the gradient, divergence, and curl operators, and the fundamental theorems of vector calculus, such as the gradient theorem and the divergence theorem.

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