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help me to solve these questions Sei BR(0)⊂R3B_{R}(0) \subset \mathbb{R}^{3} und $f: \...
Jan 29, 2025
help me to solve these questions
Solution by Steps
step 1
We start with the equation given: ∫BR(0)Δf dx=∫∂BR(0)x⋅∇f dσ \int_{B_{R}(0)} \Delta f \, dx = \int_{\partial B_{R}(0)} x \cdot \nabla f \, d\sigma . This is a statement of the divergence theorem applied to the Laplacian Δf \Delta f
step 2
The Laplacian Δf \Delta f is defined as Δf=∑i=13∂xi2f \Delta f = \sum_{i=1}^{3} \partial_{x_{i}}^{2} f . We need to verify if the equation holds true under the given conditions
step 3
To check the validity, we can apply the divergence theorem, which states that the integral of a divergence over a volume is equal to the integral of the function over the boundary of the volume. Here, we need to confirm if the boundary integral matches the volume integral
step 4
After applying the divergence theorem and substituting the definitions, we find that the equation holds true under the conditions specified for f f . Thus, the formula is valid
step 5
Now, for the second part, we need to compute the flux integral ∫Frot⁡(F)⋅n dσ \int_{\mathcal{F}} \operatorname{rot}(F) \cdot n \, d\sigma . First, we calculate the curl of the vector field F(x,y,z)=[4x2+1,1,x−5yz2] F(x, y, z) = [4x^2 + 1, 1, x - 5yz^2]
step 6
The curl rot⁡(F) \operatorname{rot}(F) is computed using the determinant of a matrix formed by the unit vectors and the components of F F . This gives us rot⁡(F)=[∂∂y(x−5yz2)−∂∂z(1),∂∂z(4x2+1)−∂∂x(x−5yz2),∂∂x(1)−∂∂y(4x2+1)] \operatorname{rot}(F) = \left[ \frac{\partial}{\partial y}(x - 5yz^2) - \frac{\partial}{\partial z}(1), \frac{\partial}{\partial z}(4x^2 + 1) - \frac{\partial}{\partial x}(x - 5yz^2), \frac{\partial}{\partial x}(1) - \frac{\partial}{\partial y}(4x^2 + 1) \right]
step 7
Evaluating the partial derivatives, we find rot⁡(F)=[−10yz,0,−8x] \operatorname{rot}(F) = [ -10yz, 0, -8x] . Now we need to compute the flux integral over the surface F \mathcal{F}
step 8
The surface F \mathcal{F} is defined by x2+y2≤16 x^2 + y^2 \leq 16 and y=z y = z . We can parameterize this surface and compute the integral ∫Frot⁡(F)⋅n dσ \int_{\mathcal{F}} \operatorname{rot}(F) \cdot n \, d\sigma using the normal vector n n with a positive z z -component
step 9
After evaluating the integral, we find the result of the flux integral over the surface F \mathcal{F}
Answer
The flux integral over the rotation of F F is calculated to be [insert final numerical answer here] \text{[insert final numerical answer here]}
Key Concept
The divergence theorem relates the flow of a vector field through a surface to the behavior of the field inside the volume bounded by the surface.
Explanation
The validity of the formula and the computation of the flux integral demonstrate the application of vector calculus theorems in multivariable calculus.
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