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Questions and Answers
What is the first step in calculating the surface integral of the vector field 𝑉 over the sides of a cubical box?
What is the first step in calculating the surface integral of the vector field 𝑉 over the sides of a cubical box?
Which component of the vector field 𝑉 contributes to the surface integral over the face in the yz-plane where x = 0?
Which component of the vector field 𝑉 contributes to the surface integral over the face in the yz-plane where x = 0?
What is the correct orientation for the positive direction of surface integrals over the five sides of the cube?
What is the correct orientation for the positive direction of surface integrals over the five sides of the cube?
When calculating the surface integral over the face in the xz-plane where y = 1, which expression should be substituted into the vector field 𝑉?
When calculating the surface integral over the face in the xz-plane where y = 1, which expression should be substituted into the vector field 𝑉?
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What assumption is made when taking the 'positive' outward direction in the calculation of the surface integral?
What assumption is made when taking the 'positive' outward direction in the calculation of the surface integral?
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Study Notes
Surface Integral Calculation of Vector Field
- The integral is calculated over the surface of a cubical box, excluding the bottom face.
- The vector field is defined as V = (2xz, (x + 2)y, y(z - 3)).
- The direction for integration is specified as "upward and outward," meaning the normals of the surfaces will point away from the box.
Procedure for Calculation
- Divide the cube into five sides: front, back, left, right, and top.
- Calculate the integral for each surface separately using the formula for the surface integral of a vector field, which is:
- ∫∫_S V·dS
Key Faces of the Cube to Consider
-
Front Face (z = 0):
- Boundaries defined by y and x.
- Evaluate the normal vector pointing outward, which is in the direction of the positive z-axis.
-
Back Face (z = h):
- Again, define boundaries with respect to x and y.
- Use the normal vector going in the negative z direction for integration.
-
Left Face (x = 0):
- The normal vector points towards the negative x-axis.
-
Right Face (x = h):
- Evaluates the positive x direction for the surface integral.
-
Top Face (y = h):
- Positioning the normal vector in the positive y direction for surface integration.
Considerations
- Ensure to account for the appropriate limits for each side.
- Verify boundary conditions for the calculation of integrals from 0 to h, depending on the face being calculated.
Final Calculation
- Sum the contributions from all five integral calculations to find the total surface integral over the cube, ensuring all signs and directions are appropriately taken into account.
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Description
This quiz focuses on calculating the surface integral of the given vector field over the five sides of a cubical box. Students are required to approach the problem side by side, considering the positive direction for upward and outward movement. Use the hint provided to examine one side at a time for effective evaluation.
