Vector Equations and Particle Motion

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Questions and Answers

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?

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)?

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?

2, -3, 4 (A) Signup and view all the answers

What is the expression for the gradient of φ if φ = xyz?

yz î + xz ĵ + xy k̂ (D) Signup and view all the answers

At the point (1,-1,2), what is the value of grad φ when φ = xyz?

-2î + 2ĵ - k̂ (A) Signup and view all the answers

How do you calculate the magnitude of the gradient |∇φ| at the point (2,-1,1) for φ = x² + y² + z² + 2xyz?

√(2^2 + 2^2 + (-2)^2) (B) Signup and view all the answers

What is the gradient expression for φ = x² + y² + z² + 2xyz?

(2x + 2yz)î + (2y + 2xz)ĵ + (2z + 2xy)k̂ (B) Signup and view all the answers

What remains constant in the expression of the gradient ∇φ when φ = 2x³y²z⁴?

The powers of x, y, and z. (D) Signup and view all the answers

What will be the divergence of the vector field F~ = ∇φ at the point (1,-1,1) for φ = 2x³y²z⁴?

24 (B) Signup and view all the answers

What is the expression for the divergence of the vector field F~?

12xy z + 4x z + 24x y z (A) Signup and view all the answers

What is the curl of the vector field F~ = ∇φ at the point (1,-1,1) for φ = 2x³y²z⁴?

(0, 0, 0) (B) Signup and view all the answers

What does the curl of the vector field F~ equal at any point?

0 (D) Signup and view all the answers

Which of the following is true about the gradient vector field?

It points in the direction of the maximum rate of increase of the scalar field. (C) Signup and view all the answers

At the point (1,-1,1), what is the calculated divergence of F~?

40 (C) Signup and view all the answers

Which term is not part of the expression for the curl of F~?

8x3 yz4 (D) Signup and view all the answers

How is the divergence of vector field F~ computed?

By summing the first partial derivatives of the components. (B) Signup and view all the answers

What is the coefficient of the î component in the vector F~?

6x2 y2 z4 (B) Signup and view all the answers

Which of the following is part of the vector F~ equation?

2x3 yz4 (B) Signup and view all the answers

What is the form of the operator used to calculate divergence?

∇· (A) Signup and view all the answers

What is the computed value of div F~ at point (1,-1,1) if each component yields 12xy z, 4xz, and 24xy z respectively?

40 (D) Signup and view all the answers

What is the expression for the force field F~ given in the problem?

(2x - y + z)î + (x + y - z 2 )ĵ + (3x - 2y)k̂ (C) Signup and view all the answers

What are the parametric equations for the circle C with center at the origin and radius 3?

x = 3 cos t, y = 3 sin t, z = 0 (D) Signup and view all the answers

How is the work done in moving a particle expressed mathematically?

$W = \int_C F~ · d~r$ (C) Signup and view all the answers

What is the resulting expression for F~(r(t)) during the calculation of work?

(6 cos t - 3 sin t)î + (3 cos t + 3 sin t)ĵ + (9 cos t - 6 sin t)k̂ (B) Signup and view all the answers

What is the derivative of r(t) with respect to t?

-3 sin tî + 3 cos tĵ (C) Signup and view all the answers

How is the work calculated over the interval from 0 to $2\pi$?

$\int_0^{2\pi} (9 - 9 cos t sin t) dt$ (B) Signup and view all the answers

What is the final result for the work done after completing the calculations?

$18\pi$ units (B) Signup and view all the answers

Which trigonometric identity simplifies the expression $\int_0^{2\pi} cos t sin t dt$ in the calculations?

$\int_0^{2\pi} \frac{1}{2} sin(2t) dt$ (D) Signup and view all the answers

What aspect of the curve C is critical for defining the work done in a force field?

The orientation and smoothness of the curve (D) Signup and view all the answers

What does the term 'work' specifically refer to in this context?

The energy transferred by a force (C) Signup and view all the answers

What is the line integral of a vector function F over a curve C defined parametrically as r(t) from a to b?

$\int_a^b F(r(t)) \cdot r'(t) , dt$ (B) Signup and view all the answers

Given the vector field F = (3x^2 + 6y)î - 14yz ĵ + 20xz^2 k̂, what expression represents F evaluated at r(t)?

$9t^2 î - 14t^5 ĵ + 20t^7 k̂$ (D) Signup and view all the answers

How do you differentiate r(t) = tî + t^2 ĵ + t^3 k̂ with respect to t?

$î + 2tĵ + 3tk̂$ (D) Signup and view all the answers

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?

$9t^2 - 28t^6 + 60t^9$ (B) Signup and view all the answers

If you evaluate the definite integral of 9t^2 - 28t^6 + 60t^9 from 0 to 1, what is the result?

$5$ (C) Signup and view all the answers

From the expression $F(r(t)) = 9t^2 î - 14t^5 ĵ + 20t^7 k̂$, what is the coefficient of the ĵ component?

$-14t^5$ (B) Signup and view all the answers

What parametric equation represents a straight line segment from (x0, y0, z0) to (x1, y1, z1)?

$x = (1 - t)x_0 + tx_1$, $y = (1 - t)y_0 + ty_1$, $z = (1 - t)z_0 + tz_1$ (C) Signup and view all the answers

What is the value of vector field F evaluated at the origin (0,0,0)?

(0, 0, 0) (C), (0, 0, 0) (D) Signup and view all the answers

In the context of vector calculus, what does the differential element d~r represent?

An infinitesimal vector along the curve C in the parameterized form (C) Signup and view all the answers

What does the term piecewise smooth curve refer to in the context of line integrals?

A curve that consists of multiple smooth segments (A) Signup and view all the answers

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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Description

This quiz covers vector equations for particle motion, including position, velocity, and acceleration calculations. It also introduces the gradient of a function with examples. Test your understanding and skills in vector calculus and motion analysis.