九州大学 システム情報科学府 情報理工学専攻・電気電子工学専攻 2024年8月実施 ベクトル解析

Author

祭音Myyura (assisted by ChatGPT 5.5 Thinking)

Description

The unit vectors on , and axes of Cartesian coordinates are denoted by , and , respectively. Let the vector field

Calculate the surface integral

over the closed surface

by answering the following questions.

(1) Find the outward-pointing normal vector at each point on the closed surface .

(2) Show

on the closed surface .

(3) Find

Kai

(1)

The closed surface is given by

Define

Then the outward normal direction is given by the gradient

Since

the outward-pointing unit normal vector is

Therefore,

on .

(2)

We have

From part (1),

Hence,

Therefore,

(3)

By part (2),

By the divergence theorem,

where is the solid ellipsoid enclosed by .

Since

we have

Therefore,

Now rewrite the ellipsoid:

Dividing by , we get

Thus,

So the semi-axes are

Hence,

Since

we get

Therefore,

Hence,