n dealing with flux calculations over parametrized surfaces, especially those involving ellipsoids and vector fields, several key topics and potential pitfalls can influence the accuracy of your results. Below are summary notes covering essential areas and common areas of weakness to ensure comprehensive understanding and correct application.

    Surface Parametrization
    a. Ellipsoid Standard Form

    Equation: ( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 )
    Semi-Axes: ( a, b, c ) represent the lengths of the axes along the ( x ), ( y ), and ( z ) directions, respectively.

b. Modified Spherical Coordinates

Purpose: Adapt spherical coordinates to fit the scaled dimensions of an ellipsoid.
Parametrization:
[
\begin{cases}
x = a \sin\theta \cos\phi \
y = b \sin\theta \sin\phi \
z = c \cos\theta
\end{cases}
]
Parameter Domains:
    ( \theta \in [0, \pi] ) (polar angle)
    ( \phi \in [0, 2\pi) ) (azimuthal angle)
Adjustments Needed: Modify ( \theta ) and ( \phi ) ranges based on specific surface constraints (e.g., ( z \geq 0 ), ( y \leq 0 )).

c. Common Pitfalls

Incorrect Scaling: Forgetting to scale ( \sin\theta ) and ( \cos\theta ) by ( a, b, c ) respectively.
Improper Domain Restrictions: Not accurately restricting ( \theta ) and ( \phi ) to meet surface conditions.

    Orientation of the Surface Normal (( \mathbf{n} ))
    a. Outward vs. Inward Normal

    Outward Normal: Points away from the enclosed volume (default for closed surfaces).
    Inward Normal: Points towards the enclosed volume (required in some flux calculations).

b. Determining Orientation

Standard Parametrization: Typically yields an outward normal via ( \mathbf{r}\theta \times \mathbf{r}\phi ).
Inverting Orientation: Multiply the cross product by ( -1 ) to obtain an inward normal:
[
d\mathbf{S} = -(\mathbf{r}\theta \times \mathbf{r}\phi) , d\theta , d\phi
]

c. Common Pitfalls

Sign Errors: Incorrectly assigning the direction of the normal vector, leading to wrong flux signs.
Miscalculating Cross Products: Errors in computing ( \mathbf{r}\theta \times \mathbf{r}\phi ) affect the normal's direction and magnitude.

    Differential Surface Element (( d\mathbf{S} ))
    a. Computing ( d\mathbf{S} )

    Formula:
    [
    d\mathbf{S} = \mathbf{n} , dS = (\mathbf{r}\theta \times \mathbf{r}\phi) , d\theta , d\phi
    ]
    Components Breakdown: Calculate each component of the cross product carefully, considering the scaling factors.

b. Magnitude Considerations

Unit Normal: Ensure the normal vector is a unit vector if required by the problem.
Orientation Impact: The direction of ( \mathbf{n} ) directly influences the flux calculation.

c. Common Pitfalls

Incorrect Cross Product Calculation: Missing terms or incorrect coefficients disrupt ( d\mathbf{S} ).
Not Accounting for Scaling: Failure to include ellipsoid scaling factors affects the magnitude.

    Computing the Dot Product (( \mathbf{F} \cdot d\mathbf{S} ))
    a. Substituting Parametrization into ( \mathbf{F} )

    Process:
    Replace ( x, y, z ) in ( \mathbf{F}(x, y, z) ) with their parametrized expressions.

b. Performing the Dot Product

Formula:
[
\mathbf{F} \cdot d\mathbf{S} = F_x (dS_x) + F_y (dS_y) + F_z (dS_z)
]
Simplification: Combine like terms and simplify the resulting expression.

c. Common Pitfalls

Misplaced Terms: Incorrectly matching components during substitution.
Sign Errors: Especially when handling inward normals, leading to incorrect overall signs.

    Setting Up and Evaluating the Integral
    a. Integral Setup

    Flux Integral:
    [
    \iint_S \mathbf{F} \cdot d\mathbf{S} = \int_{\theta_1}^{\theta_2} \int_{\phi_1}^{\phi_2} (\mathbf{F} \cdot d\mathbf{S}) , d\phi , d\theta
    ]
    Limits of Integration: Based on the parameter domains established earlier.

b. Evaluating Trigonometric Integrals

Techniques:
    Substitution: Simplify integrals using substitution methods.
    Trigonometric Identities: Utilize identities to simplify integrands.
    Symmetry Considerations: Recognize when integrals cancel out or reinforce due to symmetry.

c. Common Pitfalls

Incorrect Limits: Misapplying parameter bounds alters the integration region.
Integration Errors: Mistakes in calculating antiderivatives or applying limits.
Overlooking Zero Contributions: Assuming terms vanish without verifying integral evaluations.

    Divergence Theorem vs. Direct Flux Calculation
    a. Divergence Theorem Overview

    Statement:
    [
    \iint_{\partial V} \mathbf{F} \cdot d\mathbf{S} = \iiint_V (\nabla \cdot \mathbf{F}) , dV
    ]
    Applicability: Requires a closed surface; relates flux through the surface to the volume integral of the divergence.

b. When to Use

Closed Surfaces: Beneficial for simplifying flux calculations over closed boundaries.
Non-Closed Surfaces: Not directly applicable unless the surface is closed by adding auxiliary boundaries.

c. Common Pitfalls

Misapplying to Open Surfaces: Applying the theorem without ensuring the surface is closed leads to incorrect results.
Ignoring Boundary Conditions: Failing to account for additional surfaces needed to close the region can invalidate the theorem's application.

    Verification and Comparison of Results
    a. Cross-Checking Calculations

    Parameter Consistency: Ensure parametrizations match the surface's geometry and constraints.
    Component Matching: Verify each component in ( \mathbf{F} ) and ( d\mathbf{S} ) is correctly accounted for in the dot product.

b. Intermediate Step Validation

Partial Results: Compare intermediate expressions for ( \mathbf{F} \cdot d\mathbf{S} ) and integral terms between different approaches.
Symmetry Exploitation: Use symmetrical properties of the surface or vector field to simplify and verify calculations.

c. Common Pitfalls

Overlooking Negative Signs: Especially in flux direction and orientation.
Incomplete Integrals: Missing contributions from parts of the surface or integral bounds.

Areas of Potential Weakness and How to Address Them

Orientation Handling:
    Weakness: Confusion between inward and outward normals.
    Solution: Carefully determine the required normal orientation and adjust the cross product accordingly. Visual aids or diagrams can help.

Parametrization Accuracy:
    Weakness: Incorrect scaling or domain restrictions leading to improper surface representation.
    Solution: Double-check parametrization formulas and ensure the parameter domains correctly reflect surface constraints.

Cross Product Calculations:
    Weakness: Errors in computing ( \mathbf{r}\theta \times \mathbf{r}\phi ).
    Solution: Methodically compute each component of the cross product, possibly using determinant expansion step-by-step.

Integration Techniques:
    Weakness: Mistakes in evaluating trigonometric integrals or applying substitution methods.
    Solution: Practice integral techniques, verify antiderivatives, and ensure correct application of limits.

Sign Conventions:
    Weakness: Incorrect signs in ( d\mathbf{S} ) leading to inverted flux values.
    Solution: Consistently track sign changes, especially when adjusting for normal orientation.

Theorem Misapplication:
    Weakness: Applying the Divergence Theorem to open surfaces or without considering necessary boundary conditions.
    Solution: Ensure the surface is closed before applying the theorem or opt for direct flux calculations when dealing with open surfaces.

Component Substitutions:
    Weakness: Incorrectly substituting parametrized variables into ( \mathbf{F} ).
    Solution: Carefully replace ( x, y, z ) with their parametrized expressions and verify each substitution step.

Boundary Contributions:
    Weakness: Missing contributions from parts of the surface not covered by parametrization.
    Solution: Ensure the parametrization fully describes the intended surface region and check for additional boundary surfaces if needed.

Final Recommendations

Systematic Approach: Follow a step-by-step procedure—parametrize the surface, compute ( d\mathbf{S} ), substitute into ( \mathbf{F} ), perform the dot product, and then integrate.
Peer Review: Have someone else review your steps to catch unnoticed errors.
Use Software Tools: Utilize mathematical software (e.g., MATLAB, Mathematica) to verify cross products and integrals.
Understand Theorems Deeply: Grasp the foundational concepts behind theorems like the Divergence Theorem to apply them correctly.
Practice Similar Problems: Reinforce understanding by solving various flux calculation problems with different surfaces and vector fields.

By thoroughly understanding and addressing these areas, you can enhance the accuracy of your flux calculations and effectively troubleshoot discrepancies in your results.
2. Let S be the part of the ellipsoid 4x2 + 9y2 + z2 = 1, with z ≥ 0, y ≤ 0.
(a) Find a parametrisation of S by modifying the spherical coordinates.
3
(b) Find the tangent plane to the surface S at 0, −
√3
6 , 1
2
!
. (Show your work.)
(c) Let
F(x, y, z) = 3yi + 2zj + 8x2k.
Orient the surface S using the inward unit normal. Using your parametrisation
in (a), evaluate the flux Z Z
S
F · dS.

Solve problem 2 parts A,b, and C making sure to show working out in full and also make sure that you pay attention to the notes that are given above, as they cover some pitfalls encountered by students when solving the problem. Write the steps for the working out in full without fail.