Vector Fields

Problem 1: Identification of a Vector Field

• Task: Identify the vector field plotted alongside the given problem statement.

Problem 2: Plot the Vector Field

• Given Vector Field: ( extbf{F} = \langle -y, x \rangle )

Gradient Vectors

Problem 3: Gradient Vector Calculation

• Task: Find the gradient vector of the function ( f(x, y) = 4x^2 \sin(y) + e^x )
• Definition of Gradient Vector: The gradient of a scalar function ( f ) is defined as:   - ( \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) )
• Steps to Calculate:   - Compute ( \frac{\partial f}{\partial x} ) which yields ( 8x \sin(y) + e^x )   - Compute ( \frac{\partial f}{\partial y} ) which yields ( 4x^2 \cos(y) )
• Resulting Gradient Vector: ( \nabla f = \left( 8x \sin(y) + e^x, 4x^2 \cos(y) \right) )

Problem 4: Conservativeness of a Vector Field

• Given Vector Field: ( extbf{F} = \langle x^2 y^2, x^2 y^2 \rangle )
• Definition: A vector field ( \textbf{F} = \langle f, g \rangle ) in ( \mathbb{R}^2 ) is conservative if:   - ( \frac{\partial g}{\partial x} = \frac{\partial f}{\partial y} )
• Determination Steps:   - Compute ( \frac{\partial g}{\partial x} = \frac{\partial (x^2 y^2)}{\partial x} = 2xy^2 )   - Compute ( \frac{\partial f}{\partial y} = \frac{\partial (x^2 y^2)}{\partial y} = 2x^2y )
• Conclusion: The vector field is conservative.
• Finding Potential Function: If ( \textbf{F} ) is conservative, find ( f(x, y) ) such that ( \nabla f = \textbf{F} ).

Examination of Additional Vector Fields

Problem 5: Finding Potential Function

• Given Vector Field: ( extbf{F} = \langle x^2 \arctan(y) + 2\cos(2x), y^2 + x^3 + 3y \rangle )
• Determine Conservativeness: Apply conditions from previous definition.

Problem 6: Finding Potential Function

• Given Vector Field: ( extbf{F} = \langle 3x^2 z + \frac{1}{x}, 2yz^2 + 3e^x, x^2 + 2y^2 z \rangle )
• Similar to Problem 5, analyze for conservativeness.

Divergence and Curl

Definitions

• Divergence: ( \text{div} \textbf{F} = \nabla \cdot \textbf{F} )
• Curl: ( \text{curl} \textbf{F} = \nabla \times \textbf{F} )

Problem 7: Divergence Calculation

• Given Vector Field: ( extbf{F} = \langle 2y, 0 \rangle )
• Compute Divergence: ( \text{div} \textbf{F} = \frac{\partial (2y)}{\partial x} + \frac{\partial (0)}{\partial y} = 0 )

Problem 8: Curl Calculation

• Given Vector Field: ( extbf{F} = \langle 2y, 0 \rangle )
• Compute Curl: ( \text{curl} \textbf{F} = 0 ) (as the vector field has no rotation).

Problem 9: Divergence and Curl Evaluation

• Given Vector Field: ( extbf{F} = \langle -2y, 2x \rangle )
• Compute:   - Divergence: ( \text{div} \textbf{F} = -2 + 2 = 0 )   - Curl: ( \text{curl} \textbf{F} = 0 )

Problem 10: Divergence and Curl Evaluation

• Given Vector Field: ( extbf{F}(x, y) = \langle 2x, 2y \rangle )
• Compute:   - Divergence: ( \text{div} \textbf{F} = 2 + 2 = 4 )   - Curl: ( \text{curl} \textbf{F} = 0 )

Problem 11: Divergence and Curl Evaluation

• Given Vector Field: ( extbf{F} = \langle x^2y, 2y + x, 4xz + 3y \rangle )
• To be calculated:   - Divergence: ( \text{div} \textbf{F} = \frac{\partial (x^2 y)}{\partial x} + \frac{\partial (2y + x)}{\partial y} + \frac{\partial (4xz + 3y)}{\partial z} )   - Curl: Use the formula for curl.

Parameterization

Problem 12: Parameterization of a Parabola

• Goal: Parameterize the curve segment defined by ( y = x^2 ) from coordinate ( (1, 1) ) to ( (2, 4) ).
• Suggested Parameterization: Let ( x = t ), then ( y = t^2 ) with bounds ( 1 \leq t \leq 2 ).

Problem 13: Parameterization of Intersection of Surfaces

• Surfaces:   - Sphere: ( x^2 + y^2 = 9 )   - Plane: ( 2x + 3y + z = 8 )
• To find the parameterization: Use trigonometric functions to represent the circle defined by the sphere, and solve for ( z ).

Line Integrals Over Scalar Fields

Steps to Evaluate Line Integrals

1. Parameterize your curve. Ensure to include bounds for the parameterization.
2. Compute the length element: ( ds = |\mathbf{r}'(t)| dt ).
3. Substitute the parameterization into the integral:    - Plug in for ( x, y, z ) and replace accordingly.
4. Integrate under specified bounds.

Problem 14: Evaluate the Line Integral

• Integral: Evaluate ( \int_C 2xy^2 ds ) where C is the right half of the circle ( x^2 + y^2 = 9 ).

Problem 15: Compute the Integral

• Integral: ( \int_C x \cos(y) ds ) where C is defined as the piece of the curve ( x = \sin(y) ) from point ( (0, 0) ) to point ( (1, 1) ).

Line Integrals Over Vector Fields

Evaluation of Vector Field Integrals

1. Parameterize your curve. Remember to add bounds.
2. Compute ( \mathbf{r}'(t) ) to reflect the derivative of the parameterization.
3. Substitute coordinates into the integral formula:    - Use: ( d\mathbf{r} = \mathbf{r}'(t) dt )
4. Integrate over bounds determined by the parameterization.

Problem 16: Evaluate Scalar Field Dot Product

• Evaluating: ( \int_C \textbf{F} \cdot d\mathbf{r} ) for ( extbf{F} = \langle 2y, x^2 \rangle ) over the segment from ( (1, 2) ) to ( (-1, 3) ).

Problem 17: Evaluate Helix Dot Product

• Evaluating: ( \int_C \textbf{F} \cdot d\mathbf{r} ) for ( extbf{F} = \langle 2y, 3xz, x^2 \rangle ) over the helix defined by ( extbf{r}(t) = \langle t, \sin(t), \cos(t) \rangle ) from ( t = 0 ) to ( t = 2\pi ).

Applications of Line Integrals

Problem 18: Evaluate along Curve

• Evaluate: ( \int x^2y \, dx + 2y \, dy ) along the curve ( y = x^2 ) from ( (0, 0) ) to ( (3, 9) ).

Problem 19: Evaluate along Curve

• Evaluate: ( \int x^2y \, dx + 2y \, dy ) along the curve ( y = 3x ) from ( (0, 0) ) to ( (3, 9) ).

Fundamental Theorem of Line Integrals

• Connection: The fundamental theorem relates line integrals of conservative vector fields along a path to the values of a potential function at the endpoints of the path.

Problem 20: Further Evaluation Along Curve

• Evaluate: ( \int 3x^2y \, dx + x^2 \, dy ) along the curve ( y = x^2 ) from ( (0, 0) ) to ( (3, 9) ).

Problem 21: Vector Field Dot Product Evaluation

• Compute: ( \int_C extbf{F} \cdot d\mathbf{r} ) for ( extbf{F} = \langle 3x^2z + 1, 2yz^2 + 3e^x, x^2 + 2y^2z \rangle ) where C is the curve defined by ( extbf{r}(t) = \langle 3t^2 + 1, 2t, t^2 + 1 \rangle ) for bounds of ( t = 0 ) to ( t = 1 ).

Problem 22: Elliptical Path Evaluation

• Evaluate: ( \int_C extbf{F} \cdot d\mathbf{r} ) where ( extbf{F} = \langle 3y^2e^{x^2} + 2x^2, 3ye^{x^2} \rangle ) along one turn of the ellipse ( 4x^2 + 9y^2 = 144 ).

Problem 23: Intersection Surface Evaluation

• Evaluate: ( \int_C extbf{F} \cdot d\mathbf{r} ) for ( extbf{F} = \langle y^2, 2xy + \cos(y) \rangle ) along the intersection of surfaces defined by ( x^2 + y^2 + z^2 = 5 ) and ( z = 3x^2 + 3y^2 + 2 ).