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Questions and Answers
How would you approach calculating the surface integral of the given vector field over the five sides of a cubical box?
I would calculate the surface integral for each side of the cube separately, applying the outward normal vector and integrating the vector field accordingly.
What is the divergence theorem, and how does it apply to checking the function V over the unit cube?
The divergence theorem states that the surface integral of a vector field over a closed surface is equal to the volume integral of the divergence of the field. To check the function V, I would compute the divergence and then integrate it over the volume of the unit cube.
How do you find the unit normal vector to the surface of an ellipsoid defined by the equation u = x^2 + y^2 + z^2?
To find the unit normal vector, I would take the gradient of the ellipsoid equation, then normalize it by dividing by its magnitude.
What steps would you follow to calculate the line integral of the vector A around a closed path? 𝑉"⃗ =(2𝑥𝑧 )i+ (𝑥 + 2)j + 𝑦(𝑧^2 − 3)𝑧̂
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What is the relationship between the line integral of vector A and the surface integral of its curl over the enclosed area?
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Study Notes
Assignment Overview
- Assignment for PHYS 311 carries a total weight of 35%.
- Due date is set for September 5, 2024, during class.
Surface Integral Calculation
- Task involves calculating the surface integral of a vector field ( \mathbf{V} = 2xz \hat{i} + (x + 2)y \hat{j} + y(z - 3) \hat{k} ).
- Calculation excludes the bottom side of a specified cubical box.
- Positive direction for integration is defined as "upward and outward" in correspondence with given arrows.
- Suggested method: Evaluate one side of the cube at a time.
Divergence Theorem Validation
- Validate the divergence theorem using the vector function ( \mathbf{V} = y^2 \hat{i} + (2xy + z^2) \hat{j} + (2yz) \hat{k} ).
- Focus on the unit cube situated at the origin for the application of the theorem.
Ellipsoid Normal Vectors
- Equation for a family of ellipsoids is given by ( u = a^2 + b^2 + c^2 ).
- Task is to determine the unit vector normal to each point on the surface of these ellipsoids.
- Hint provided: Find the unit vector ( \mathbf{n} ).
Line Integral and Surface Integral
- Given vector ( \mathbf{A} = 4\hat{r} + 3\hat{\theta} - 2\hat{\phi} ) requires evaluation of its line integral around a specified closed path.
- The closed path includes a circular arc of radius ( r_0 ), centered at the origin.
- Also required: Calculate the surface integral of ( \nabla \times \mathbf{A} ) over the area enclosed by the path.
- Comparison of results from line integral and surface integral is necessary.
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Description
This quiz focuses on the surface integral calculation of a given vector field and validating the divergence theorem using a specified vector function. Additionally, it involves finding the unit normal vectors for ellipsoids. It is suitable for students of PHYS 311, enhancing their understanding of vector calculus concepts.
