Explanation Given: The given equation is z = x 2 + 2 y 2 . Calculation: As the given equation z = x 2 + 2 y 2 . where x = r cos θ and y = r sin θ . Therefore, y x = r sin θ r cos θ = tan θ ...................... ( 1 ) Since, z = x 2 + 2 y 2 ................................. ( 2 ) Substitute eq (1) for y into eq (2), z = x 2 + 2 y 2 z = ( y tan θ ) 2 + 2 y 2 = ( 1 tan 2 θ + 2 ) y 2 = ( cot 2 θ + 2 ) y 2 ( ∂ z ∂ y ) θ = ∂ ( ( cot 2 θ + 2 ) y 2 ) ∂ x | θ ( ∂ z ∂ y ) θ = 2 y ( cot 2 θ + 2 ) Now, repeat z = r 2 tan 2 θ , where x = r cos θ and y = r sin θ . Therefore, x 2 + y 2 = r 2 ( cos 2 θ + sin 2 θ ) = r 2 ............. ( 1 ) y x = r sin θ r cos θ = tan θ

Mathematical Methods in the Physical Sciences

3rd Edition

ISBN: 9780471198260

Author: Mary L. Boas

Publisher: Wiley, John & Sons, Incorporated

See similar textbooks

Question

Chapter 4.1, Problem 12P

To determine

Tofind:the value of ( ∂z ∂y)θ at the given points in the given equation.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.1, Problem 11P

Chapter 4.1, Problem 13P

Students have asked these similar questions

The Martin-Beck Company operates a plant in St. Louis with an annual capacity of 30,000 units. Product is shipped to regional distribution centers located in Boston, Atlanta, and Houston. Because of an anticipated increase in demand, Martin-Beck plans to increase capacity by constructing a new plant in one or more of the following cities: Detroit, Toledo, Denver, or Kansas. The following is a linear program used to determine which cities Martin-Beck should construct a plant in. Let y₁ = 1 if a plant is constructed in Detroit; 0 if not y₂ = 1 if a plant is constructed in Toledo; 0 if not y₂ = 1 if a plant is constructed in Denver; 0 if not y = 1 if a plant is constructed in Kansas City; 0 if not. The variables representing the amount shipped from each plant site to each distribution center are defined just as for a transportation problem. *,, = the units shipped in thousands from plant i to distribution center j i = 1 (Detroit), 2 (Toledo), 3 (Denver), 4 (Kansas City), 5 (St.Louis) and…

Consider the following mixed-integer linear program. Max 3x1 + 4x2 s.t. 4x1 + 7x2 ≤ 28 8x1 + 5x2 ≤ 40 x1, x2 ≥ and x1 integer (a) Graph the constraints for this problem. Indicate on your graph all feasible mixed-integer solutions. On the coordinate plane the horizontal axis is labeled x1 and the vertical axis is labeled x2. A region bounded by a series of connected line segments, and several horizontal lines are on the graph. The series of line segments connect the approximate points (0, 4), (3.889, 1.778), and (5, 0). The region is above the horizontal axis, to the right of the vertical axis, and below the line segments. At each integer value between 0 and 4 on the vertical axis, a horizontal line extends out from the vertical axis to the series of connect line segments. On the coordinate plane the horizontal axis is labeled x1 and the vertical axis is labeled x2. A region bounded by a series of connected line segments, and several…

Consider the nonlinear optimization model stated below. Min s.t. 2x²-18x + 2XY + y² - 14Y + 53 x + 4Y ≤ 8 (a) Find the minimum solution to this problem. |at (X, Y) = (b) If the right-hand side of the constraint is increased from 8 to 9, how much do you expect the objective function to change? Based on the dual value on the constraint X + 4Y ≤ 8, we expect the optimal objective function value to decrease by (c) Resolve the problem with a new right-hand side of the constraint of 9. How does the actual change compare with your estimate? If we resolve the problem with a new right-hand-side of 9 the new optimal objective function value is| , so the actual change is a decrease of rather than what we expected in part (b).

Chapter 4 Solutions

Mathematical Methods in the Physical Sciences

Ch. 4.1 - If u=x2/x2+y2, find u/x,u/y.Ch. 4.1 - Prob. 2P Ch. 4.1 - If z=lnu2+v2+w2, find z/u,z/v,z/w.Ch. 4.1 - For w=x3y32xy+6, find 2w/ax2 and 2w/ay2 at the...Ch. 4.1 - For w=8x4+y42xy2, find 2w/x2 and 2w/y2 at the...Ch. 4.1 - For u=excosy, (a) verify that 2u/xy=2u/yx; (b)...Ch. 4.1 - Prob. 7P Ch. 4.1 - If z=x2+2y2,x=rcos,y=rsin, find the following...Ch. 4.1 - If z=x2+2y2,x=rcos,y=rsin, find the following...Ch. 4.1 - If z=x2+2y2,x=rcos,y=rsin, find the following...

Knowledge Booster

Tofind: the value of ( ∂ z ∂ y ) θ at the given points in the given equation.

Tofind:the value of ( ∂z ∂y)θ at the given points in the given equation.

Want to see the full answer?

Chapter 4 Solutions