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1月実施ベクトル解析

Author

祭音Myyura (assisted by ChatGPT 5.5)

(1) Prove the following relationship between the volume integral and the surface integral:

(2) Prove the following relationship between the volume integral and the surface integral:

Here,

where is the outward unit normal vector to the closed surface .

We use the divergence theorem:

Choose the vector field

Then

Using the product rule for divergence,

Since

we have

Therefore,

By the divergence theorem,

Hence,

as required.

We again use the divergence theorem.

Choose the vector field

Then

Using linearity of divergence,

From the product rule,

and

Thus,

Since the dot product is commutative,

these two terms cancel. Therefore,

Now applying the divergence theorem,

Substituting

we get

Hence,

as required.