Question

Use the surface integral in​ Stokes' Theorem to calculate the circulation of the field Bold Upper F equals x squared Bold i plus 4 x Bold j plus z squared Bold k around the curve​ C: the ellipse 16x squared plus 4 y squared equals 3 in the​ xy-plane, counterclockwise when viewed from above.

Answer 1

The question asks to calculate the circulation of a given vector field around an ellipse using Stokes' Theorem, which requires the curl of the field and a surface integral. However, details about the explicit surface the ellipse bounds are missing, and thus, the calculation cannot be completed as is.

Step-by-step explanation:

The question asks us to calculate the circulation of the vector field ℛ₀ = x²ᵂ + 4xᵀ9 + z²ᵀb around the curve C: the ellipse 16x² + 4y² = 3 in the xy-plane, counterclockwise when viewed from above. To solve this, we should use Stokes' Theorem, which relates the surface integral of the curl of the vector field over a surface S to the line integral of the vector field over the boundary curve C of the surface.

We begin by finding the curl of the vector field and then apply it to compute the surface integral. However, the question provided does not include all the necessary details to complete the calculation, such as the explicit surface that the ellipse bounds, nor the limits for integration. Without this information or further context, we cannot provide the exact value of the circulation.

In a fully worked out solution, once the surface S is defined, we would parameterize it, calculate the curl of ℛ₀, take the dot product with the unit normal vector of the surface, and integrate over S to find the circulation around C.

Answer 2

The circulation of the field f(x) over curve C is Zero

Step-by-step explanation:

The function
`f(x)=(x^(2),4x,z^(2))` and curve C is ellipse of equation

`16x^(2) + 4y^(2) = 3`

Theory: Stokes Theorem is given by:

`I= \int \int\limits {{Curl f\cdot \hat{N }} \, dx`

Where, Curl f(x) =
`\left[\begin{array}{ccc}\hat{i}&\hat{j}&\hat{k}\\(∂)/(∂x) &(∂)/(∂y) &(∂)/(∂z) \\F1&F2&F3\end{array}\right]`

Also, f(x) = (F1,F2,F3)

`\hat{N} = grad(g(x))`

Using Stokes Theorem,

Surface is given by g(x) =
`16x^(2) + 4y^(2) - 3`

Therefore, tex]\hat{N} = grad(g(x))[/tex]

`\hat{N} = grad(16x^(2) + 4y^(2) - 3)`

`\hat{N} = (32x,8y,0)`

Now,
`f(x)=(x^(2),4x,z^(2))`

Curl f(x) =
`\left[\begin{array}{ccc}\hat{i}&\hat{j}&\hat{k}\\(∂)/(∂x) &(∂)/(∂y) &(∂)/(∂z) \\F1&F2&F3\end{array}\right]`

Curl f(x) =
`\left[\begin{array}{ccc}\hat{i}&\hat{j}&\hat{k}\\(∂)/(∂x) &(∂)/(∂y) &(∂)/(∂z) \\x^(2)&4x&z^(2)\end{array}\right]`

Curl f(x) = (0,0,4)

Putting all values in Stokes Theorem,

`I= \int \int\limits {Curl f\cdot \hat{N} } \, dx`

`I= \int \int\limits {(0,0,4)\cdot(32x,8y,0)} \, dx`

`I= \int \int\limits {(0,0,4)\cdot(32x,8y,0)} \, dx`

I=0

Thus, The circulation of the field f(x) over curve C is Zero