Use the Divergence Theorem to calculate the surface integral
S
F · dS;
that is, calculate the flux of F across S.
F(x, y, z) = x^4i − x^3z^2j + 4xy^2zk,
S is the surface of the solid bounded by the cylinder
x^2 + y^2 = 4
and the planes
z = x + 6 and z = 0.
To use the Divergence Theorem to calculate the surface integral of a vector field across a closed surface, we need to follow these steps:
1. Identify the closed surface bounded by the given solid. In this case, the closed surface S is the surface of the solid bounded by the cylinder x^2 + y^2 = 4 and the planes z = x + 6 and z = 0.
2. Determine the orientation of the surface. The orientation is important because it affects the sign of the surface integral. You can choose any consistent orientation as long as you are consistent throughout the calculations.
3. Calculate the divergence of the vector field F. The divergence of a vector field F in three-dimensional space is calculated as follows: div(F) = ∇ · F = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z, where ∇ is the vector differential operator (del operator).
In this case, F(x, y, z) = x^4i - x^3z^2j + 4xy^2zk, so we need to calculate ∂Fx/∂x, ∂Fy/∂y, and ∂Fz/∂z.
∂Fx/∂x = 4x^3
∂Fy/∂y = 0 (since there is no y component in the vector field)
∂Fz/∂z = 4xy^2
Therefore, div(F) = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z = 4x^3 + 4xy^2.
4. Apply the Divergence Theorem. The Divergence Theorem states that the surface integral of a vector field across a closed surface is equal to the triple integral of the divergence of the vector field over the volume enclosed by the surface.
Mathematically, the surface integral ∮S F · dS is equal to ∭V div(F) dV, where V is the volume enclosed by the surface.
Since the surface S is defined by a cylinder and two planes, we can express the volume V as V = {(x, y, z) | 0 ≤ z ≤ x + 6, x^2 + y^2 ≤ 4}.
Now we can evaluate the triple integral ∭V div(F) dV.
5. After performing the triple integral, the result will give you the flux of the vector field F across the surface S.
I hope this explanation helps you understand the process of using the Divergence Theorem to calculate the given surface integral. Let me know if you need further assistance with the calculations.
To calculate the surface integral using the Divergence Theorem, we need to follow these steps:
Step 1: Find the divergence of the vector field F.
Step 2: Determine the volume enclosed by the surface S.
Step 3: Apply the Divergence Theorem to calculate the surface integral.
Let's proceed with each step.
Step 1: Find the divergence of the vector field F.
The divergence of a vector field F = (P, Q, R) is given by:
div(F) = ∂P/∂x + ∂Q/∂y + ∂R/∂z.
In our case, F(x, y, z) = x^4i - x^3z^2j + 4xy^2zk, so:
∂P/∂x = ∂/∂x (x^4) = 4x^3,
∂Q/∂y = ∂/∂y (-x^3z^2) = 0,
∂R/∂z = ∂/∂z (4xy^2) = 0.
Therefore, the divergence of F is:
div(F) = 4x^3.
Step 2: Determine the volume enclosed by the surface S.
The surface S is bounded by the cylinder x^2 + y^2 = 4 and the planes z = x + 6 and z = 0.
To determine the volume enclosed by the surface S, we need to find the limits of integration. In this case, since the cylinder is symmetric about the z-axis, we can use cylindrical coordinates.
The limits for the cylindrical coordinates are:
- r: 0 to 2 (due to x^2 + y^2 = 4)
- θ: 0 to 2π (due to the cylindrical symmetry)
- z: 0 to x + 6
Step 3: Apply the Divergence Theorem to calculate the surface integral.
The Divergence Theorem states that the flux of a vector field F across a closed surface S is equal to the triple integral of the divergence of F over the volume enclosed by S.
The surface integral can be calculated as:
∬S F · dS = ∭V div(F) dV,
where ∭V represents the triple integral over the volume V enclosed by the surface S.
In our case, since the divergence is 4x^3, the surface integral becomes:
∬S F · dS = ∭V (4x^3) dV.
Now we can evaluate the triple integral using the limits of integration determined in step 2.
∬S F · dS = ∫₀² ∫₀²π ∫₀^(x+6) (4x^3) dz dθ dr.
Evaluating this integral will give us the final result of the surface integral.