Compute the Area of the Region Outside the Circle r = 3 and Outside the Cardioid: Multivariable Calculus Assignment, CU, Canada

University Carleton University (CU) Canada
Subject Multivariable Calculus

Assignment Questions:

1. Compute the area of the region outside the circle r = 3 and outside the cardioid r = 2(1 + cos(θ)).

2. (a) Please kindly find the length of the curve defined by the expression r(θ) = k sec(θ), θ ∈ [0, β]; k is a constant. (b) Sketch the curve for k = 1 and β = π/3.

3. In polar coordinates the position of a point-particle is described by the vector ~r = rrˆ(θ), where r = r(t) is the radial coordinate, rˆ(θ) is a unit vector in the radial direction, θ = θ(t). The parameter t represents time. Knowing that (d/ dθ) rˆ= θ ˆ, (d/dθ)θˆ = −r ˆ, where θˆ is a unit vector in the angular direction, use the chain rule to show that the velocity and acceleration vectors are, respectively,

Multivariable Calculus

Notation: the prime denotes a derivative with respect to time.
Remark: vr = r′ is the tangencial velocity; vθ = rθ′ is the angular velocity; aθ = rθ′2 is the centripetal acceleration; ac = 2r′θ′ is the coriolis acceleration.

Take professional academic assistance & Get 100% Plagiarism free papers

4. The ideal gas equation for one mol reads P V = RT, where the universal gas constant is R = 8.314 J/mol/K. Find the rate at which the pressure P of the gas is changing when its volume V is 20 L and is increasing at a rate of 0.2 L/s, and its temperature T is 300 K and is increasing at a rate of 0.3 K/s. Dimensions: J=Joules, K=Kelvins, L=Litres, s=seconds. The pressure comes in Pascals.

Hint: Set P = P (V, T ) where both V and T depend on time; apply the chain rule to find the rate dP/dt, t: time.

5. The height of a hill (in feet) is given by h(x, y) = 20(16 − 4x²− 3y²+ 2xy + 28x − 18y); x denotes the distance (in miles) east and y stands for the distance (in miles) north of Bolton City. In what direction is the slope of the hill steepest at a point 1 mile north and 1 mile east of Bolton? What is the steepest slope at that point?

6. The simplest dispersive wave equation is ut + ux + uxxx = 0, u ≡ u(x, t). Show that the harmonic function u(x, t) = B sin(kx − ωt) is a solution of the above differential equation if ω = k−k³, the dispersion relation.

If you have any problem in solving your Multivariable Calculus assignment then you can seek our online assignment help in Canada at any time. We, at StudentsAssignmentHelp.com, work 24/7 hours to support university students with different calculus assignments. Our writers have extensive knowledge and in-depth experience in preparing top-notch quality assignments.

Answer
Get Support Instantly
info@studentsassignmenthelp.com
Quick Connect