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Questions and Answers
What is the expression for the velocity vector ~v of the particle at time t?
What is the expression for the velocity vector ~v of the particle at time t?
- tî - 3t^2 ĵ + 4t^3 k̂
- 2tî - 3t^2 ĵ + 4t^3 k̂ (correct)
- 2tî - 3ĵ + 4k̂
- 2tî - 3t ĵ + 4k̂
What is the acceleration vector ~a of the particle at t = 1?
What is the acceleration vector ~a of the particle at t = 1?
- 2î - 6ĵ + 12k̂ (correct)
- 2î - 3ĵ + 4k̂
- 2î - 3t^2 ĵ + 4t^3 k̂
- 2î - 6t ĵ + 12t^2 k̂
What is the formula for velocity as derived from the position vector ~r(t)?
What is the formula for velocity as derived from the position vector ~r(t)?
- d~r/dt = î + 3ĵ + 4k̂
- d~r/dt = t^2 î - t^3 ĵ + t^4 k̂
- d~r/dt = 2tî - 3t^2 ĵ + 4t^3 k̂ (correct)
- d~r/dt = î - ĵ + k̂
At time t = 1, what components make up the velocity vector ~v?
At time t = 1, what components make up the velocity vector ~v?
What is the expression for the gradient of φ if φ = xyz?
What is the expression for the gradient of φ if φ = xyz?
At the point (1,-1,2), what is the value of grad φ when φ = xyz?
At the point (1,-1,2), what is the value of grad φ when φ = xyz?
How do you calculate the magnitude of the gradient |∇φ| at the point (2,-1,1) for φ = x² + y² + z² + 2xyz?
How do you calculate the magnitude of the gradient |∇φ| at the point (2,-1,1) for φ = x² + y² + z² + 2xyz?
What is the gradient expression for φ = x² + y² + z² + 2xyz?
What is the gradient expression for φ = x² + y² + z² + 2xyz?
What remains constant in the expression of the gradient ∇φ when φ = 2x³y²z⁴?
What remains constant in the expression of the gradient ∇φ when φ = 2x³y²z⁴?
What will be the divergence of the vector field F~ = ∇φ at the point (1,-1,1) for φ = 2x³y²z⁴?
What will be the divergence of the vector field F~ = ∇φ at the point (1,-1,1) for φ = 2x³y²z⁴?
What is the expression for the divergence of the vector field F~?
What is the expression for the divergence of the vector field F~?
What is the curl of the vector field F~ = ∇φ at the point (1,-1,1) for φ = 2x³y²z⁴?
What is the curl of the vector field F~ = ∇φ at the point (1,-1,1) for φ = 2x³y²z⁴?
What does the curl of the vector field F~ equal at any point?
What does the curl of the vector field F~ equal at any point?
Which of the following is true about the gradient vector field?
Which of the following is true about the gradient vector field?
At the point (1,-1,1), what is the calculated divergence of F~?
At the point (1,-1,1), what is the calculated divergence of F~?
Which term is not part of the expression for the curl of F~?
Which term is not part of the expression for the curl of F~?
How is the divergence of vector field F~ computed?
How is the divergence of vector field F~ computed?
What is the coefficient of the î component in the vector F~?
What is the coefficient of the î component in the vector F~?
Which of the following is part of the vector F~ equation?
Which of the following is part of the vector F~ equation?
What is the form of the operator used to calculate divergence?
What is the form of the operator used to calculate divergence?
What is the computed value of div F~ at point (1,-1,1) if each component yields 12xy z, 4xz, and 24xy z respectively?
What is the computed value of div F~ at point (1,-1,1) if each component yields 12xy z, 4xz, and 24xy z respectively?
What is the expression for the force field F~ given in the problem?
What is the expression for the force field F~ given in the problem?
What are the parametric equations for the circle C with center at the origin and radius 3?
What are the parametric equations for the circle C with center at the origin and radius 3?
How is the work done in moving a particle expressed mathematically?
How is the work done in moving a particle expressed mathematically?
What is the resulting expression for F~(r(t)) during the calculation of work?
What is the resulting expression for F~(r(t)) during the calculation of work?
What is the derivative of r(t) with respect to t?
What is the derivative of r(t) with respect to t?
How is the work calculated over the interval from 0 to $2\pi$?
How is the work calculated over the interval from 0 to $2\pi$?
What is the final result for the work done after completing the calculations?
What is the final result for the work done after completing the calculations?
Which trigonometric identity simplifies the expression $\int_0^{2\pi} cos t sin t dt$ in the calculations?
Which trigonometric identity simplifies the expression $\int_0^{2\pi} cos t sin t dt$ in the calculations?
What aspect of the curve C is critical for defining the work done in a force field?
What aspect of the curve C is critical for defining the work done in a force field?
What does the term 'work' specifically refer to in this context?
What does the term 'work' specifically refer to in this context?
What is the line integral of a vector function F over a curve C defined parametrically as r(t) from a to b?
What is the line integral of a vector function F over a curve C defined parametrically as r(t) from a to b?
Given the vector field F = (3x^2 + 6y)î - 14yz ĵ + 20xz^2 k̂, what expression represents F evaluated at r(t)?
Given the vector field F = (3x^2 + 6y)î - 14yz ĵ + 20xz^2 k̂, what expression represents F evaluated at r(t)?
How do you differentiate r(t) = tî + t^2 ĵ + t^3 k̂ with respect to t?
How do you differentiate r(t) = tî + t^2 ĵ + t^3 k̂ with respect to t?
What is the dot product result of F(r(t)) with r'(t) if F(r(t)) = 9t^2 î - 14t^5 ĵ + 20t^7 k̂ and r'(t) = î + 2tĵ + 3t^2 k̂?
What is the dot product result of F(r(t)) with r'(t) if F(r(t)) = 9t^2 î - 14t^5 ĵ + 20t^7 k̂ and r'(t) = î + 2tĵ + 3t^2 k̂?
If you evaluate the definite integral of 9t^2 - 28t^6 + 60t^9 from 0 to 1, what is the result?
If you evaluate the definite integral of 9t^2 - 28t^6 + 60t^9 from 0 to 1, what is the result?
From the expression $F(r(t)) = 9t^2 î - 14t^5 ĵ + 20t^7 k̂$, what is the coefficient of the ĵ component?
From the expression $F(r(t)) = 9t^2 î - 14t^5 ĵ + 20t^7 k̂$, what is the coefficient of the ĵ component?
What parametric equation represents a straight line segment from (x0, y0, z0) to (x1, y1, z1)?
What parametric equation represents a straight line segment from (x0, y0, z0) to (x1, y1, z1)?
What is the value of vector field F evaluated at the origin (0,0,0)?
What is the value of vector field F evaluated at the origin (0,0,0)?
In the context of vector calculus, what does the differential element d~r represent?
In the context of vector calculus, what does the differential element d~r represent?
What does the term piecewise smooth curve refer to in the context of line integrals?
What does the term piecewise smooth curve refer to in the context of line integrals?
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Study Notes
Vector Equations of a Particle
- Particle moves along a curve defined by ( x = t^2 ), ( y = -t^3 ), ( z = t^4 ).
- Position vector is represented as ( \mathbf{r}(t) = t^2 \hat{i} - t^3 \hat{j} + t^4 \hat{k} ).
Velocity and Acceleration
- Velocity ( \mathbf{v} ) is calculated as ( \frac{d\mathbf{r}}{dt} = 2t \hat{i} - 3t^2 \hat{j} + 4t^3 \hat{k} ).
- Acceleration ( \mathbf{a} ) is calculated as ( \frac{d\mathbf{v}}{dt} = 2 \hat{i} - 6t \hat{j} + 12t^2 \hat{k} ).
- At ( t = 1 ), ( \mathbf{v} = 2 \hat{i} - 3 \hat{j} + 4 \hat{k} ) and ( \mathbf{a} = 2 \hat{i} - 6 \hat{j} + 12 \hat{k} ).
Example of a Moving Particle
- A particle moves with ( x = e^{-t} ), ( y = 2 \cos(3t) ), ( z = 2 \sin(3t) ).
- Requires finding velocity and acceleration at any time.
- Velocity and acceleration depend on differentiating the position equations.
Gradient of a Function
- The gradient ( \nabla \phi ) of a function ( \phi = xyz ) includes derivatives with respect to ( x ), ( y ), and ( z ).
- At the point ( (1, -1, 2) ), the gradient is ( -2 \hat{i} + 2 \hat{j} - \hat{k} ).
Magnitude of Gradient
- For ( \phi = x^2 + y^2 + z^2 + 2xyz ), the gradient is ( \nabla \phi = (2x + 2yz, 2y + 2xz, 2z + 2xy) ).
- At ( (2, -1, 1) ), the gradient evaluates to ( 2 \hat{i} + 2 \hat{j} - 2 \hat{k} ).
- Magnitude is calculated as ( |\nabla \phi| = \sqrt{2^2 + 2^2 + (-2)^2} = 2\sqrt{3} ).
Divergence and Curl
- Given ( \phi = 2x^3 y^2 z^4 ), the vector field ( \mathbf{F} = \nabla \phi ) involves partial derivatives.
- Divergence ( \text{div} \mathbf{F} ) and curl ( \text{curl} \mathbf{F} ) are also calculated at the point ( (1, -1, 1) ).
- ( \text{div} \mathbf{F} = 40 ) and ( \text{curl} \mathbf{F} = \mathbf{0} ).
Line Integral
- Line integral of vector field ( \mathbf{F} ) along curve ( C ) defined parametrically involves the differential ( d\mathbf{r} ).
- Example illustrates how to compute ( \int_C \mathbf{F} \cdot d\mathbf{r} ).
Work Done in a Force Field
- Work done to move a particle along a defined path ( C ) in a force field ( \mathbf{F} ) is given by the line integral.
- Sample calculation shows the process of applying the definitions and evaluating ( \int_C \mathbf{F} \cdot d\mathbf{r} ).
- Required work done in moving a particle once around a circle with radius 3 is ( 18\pi ) units.
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