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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?
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?
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?
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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.
