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If F = xi+yj, calculate double integral( F·n dσ) over the part of the surface z = 4−x2 −y2 that is above the (x, y) plane, by applying the divergence theorem to the volume bounded by the surface and the piece that it cuts out of the (x, y) plane. Hint: What is F · n on the (x, y) plane?
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If F = xi+yj, calculate double integral( F·n dσ) over the part of the ...
Understanding the Problem
To calculate the double integral of F · n dσ over the surface z = 4 − x² − y², we can use the divergence theorem. This theorem relates the flow of a vector field through a closed surface to the volume integral of the divergence of the field.
Vector Field and Surface Normal
- The vector field is F = xi + yj.
- The surface n is the outward normal vector. For the surface above the xy-plane, the normal vector n points outward, which means it will be in the positive z-direction (0, 0, 1) on the xy-plane.
Calculating F · n on the xy-plane
- On the xy-plane, z = 0, so the normal vector n is (0, 0, -1).
- Therefore, F · n on the xy-plane can be calculated as:
- F · n = (xi + yj) · (0, 0, -1) = 0.
Applying the Divergence Theorem
- The divergence of F (∇ · F) is calculated as:
- ∇ · F = ∂(x)/∂x + ∂(y)/∂y = 1 + 1 = 2.
- According to the divergence theorem, the surface integral can be converted to a volume integral:
Double Integral (F · n dσ) = Volume Integral (∇ · F) dV.
Volume Calculation
- The volume V is bounded by the paraboloid z = 4 − x² − y² and the xy-plane.
- The volume integral becomes:
∫∫∫ (2) dV over the volume defined by z = 0 to z = 4 − x² − y².
- Changing to polar coordinates simplifies the integration process.
Conclusion
- By employing the divergence theorem, we can evaluate the double integral efficiently through the volume integral of the divergence of F over the defined region.
To calculate the double integral of F · n dσ over the surface z = 4 − x² − y², we can use the divergence theorem. This theorem relates the flow of a vector field through a closed surface to the volume integral of the divergence of the field.
Vector Field and Surface Normal
- The vector field is F = xi + yj.
- The surface n is the outward normal vector. For the surface above the xy-plane, the normal vector n points outward, which means it will be in the positive z-direction (0, 0, 1) on the xy-plane.
Calculating F · n on the xy-plane
- On the xy-plane, z = 0, so the normal vector n is (0, 0, -1).
- Therefore, F · n on the xy-plane can be calculated as:
- F · n = (xi + yj) · (0, 0, -1) = 0.
Applying the Divergence Theorem
- The divergence of F (∇ · F) is calculated as:
- ∇ · F = ∂(x)/∂x + ∂(y)/∂y = 1 + 1 = 2.
- According to the divergence theorem, the surface integral can be converted to a volume integral:
Double Integral (F · n dσ) = Volume Integral (∇ · F) dV.
Volume Calculation
- The volume V is bounded by the paraboloid z = 4 − x² − y² and the xy-plane.
- The volume integral becomes:
∫∫∫ (2) dV over the volume defined by z = 0 to z = 4 − x² − y².
- Changing to polar coordinates simplifies the integration process.
Conclusion
- By employing the divergence theorem, we can evaluate the double integral efficiently through the volume integral of the divergence of F over the defined region.
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If F = xi+yj, calculate double integral( F·n dσ) over the part of the surface z = 4−x2 −y2 that is above the (x, y) plane, by applying the divergence theorem to the volume bounded by the surface and the piece that it cuts out of the (x, y) plane. Hint: What is F · n on the (x, y) plane?
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If F = xi+yj, calculate double integral( F·n dσ) over the part of the surface z = 4−x2 −y2 that is above the (x, y) plane, by applying the divergence theorem to the volume bounded by the surface and the piece that it cuts out of the (x, y) plane. Hint: What is F · n on the (x, y) plane? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about If F = xi+yj, calculate double integral( F·n dσ) over the part of the surface z = 4−x2 −y2 that is above the (x, y) plane, by applying the divergence theorem to the volume bounded by the surface and the piece that it cuts out of the (x, y) plane. Hint: What is F · n on the (x, y) plane? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If F = xi+yj, calculate double integral( F·n dσ) over the part of the surface z = 4−x2 −y2 that is above the (x, y) plane, by applying the divergence theorem to the volume bounded by the surface and the piece that it cuts out of the (x, y) plane. Hint: What is F · n on the (x, y) plane?.
If F = xi+yj, calculate double integral( F·n dσ) over the part of the surface z = 4−x2 −y2 that is above the (x, y) plane, by applying the divergence theorem to the volume bounded by the surface and the piece that it cuts out of the (x, y) plane. Hint: What is F · n on the (x, y) plane? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about If F = xi+yj, calculate double integral( F·n dσ) over the part of the surface z = 4−x2 −y2 that is above the (x, y) plane, by applying the divergence theorem to the volume bounded by the surface and the piece that it cuts out of the (x, y) plane. Hint: What is F · n on the (x, y) plane? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If F = xi+yj, calculate double integral( F·n dσ) over the part of the surface z = 4−x2 −y2 that is above the (x, y) plane, by applying the divergence theorem to the volume bounded by the surface and the piece that it cuts out of the (x, y) plane. Hint: What is F · n on the (x, y) plane?.
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