Relationship between a, ß, and y to make the surface integral zero:

Given field a(x, y, z) = 2axi + byj - 3yzk, we need to find the relationship between the constants a, ß, and y in order to make the surface integral of this field over a closed unit sphere zero.

1. Surface Integral:
The surface integral of a vector field over a closed surface is given by the formula:

∬S a · dS = 0

Here, a represents the vector field, dS represents the outward differential surface area vector, and S represents the closed surface.

2. Outward Differential Surface Area Vector:
For a unit sphere, the outward differential surface area vector dS can be expressed as:

dS = r̂ dA

Where r̂ is the outward unit normal vector at each point on the sphere, and dA is the differential area element on the surface.

3. Outward Unit Normal Vector:
For a unit sphere, the outward unit normal vector r̂ can be expressed as:

r̂ = (x/r)i + (y/r)j + (z/r)k

Where r = √(x^2 + y^2 + z^2) is the distance from the origin to any point on the sphere.

4. Calculation of Surface Integral:
Substituting the expressions for a, dS, and r̂ into the surface integral formula, we get:

∬S a · dS = ∬S (2axi + byj - 3yzk) · (r̂ dA)

= ∬S (2ax(x/r) + by(y/r) - 3yz(z/r)) dA

= ∬S (2ax^2/r + by^2/r - 3yz^2/r) dA

5. Simplification:
To simplify the integral, we need to express it in terms of spherical coordinates. For a unit sphere, the differential area element dA can be expressed as:

dA = r^2 sin(θ) dθ dφ

Where θ is the polar angle and φ is the azimuthal angle.

6. Expressing the Integral in Spherical Coordinates:
Substituting the expression for dA, we get:

∬S (2ax^2/r + by^2/r - 3yz^2/r) r^2 sin(θ) dθ dφ

= ∫∫ (2ax^2 sin(θ) + by^2 sin(θ) - 3yz^2 sin(θ)) dθ dφ

7. Evaluating the Integral:
To make the surface integral zero, the integrand must be zero for all values of θ and φ. Therefore, we can equate each term to zero separately.

2ax^2 sin(θ) = 0
by^2 sin(θ) = 0
-3yz^2 sin(θ) = 0

8. Relationship between a, ß, and y:
From the first equation, 2ax^2 sin(θ) = 0, we can conclude that a = 0 or x = 0.

From the second equation, by^2 sin(θ) = 0, we can conclude that ß = 0 or y = 0.