![EBK NUMERICAL METHODS FOR ENGINEERS](https://www.bartleby.com/isbn_cover_images/8220100254147/8220100254147_largeCoverImage.jpg)
Find the gradient
(a)
(b)
(c)
(a)
![Check Mark](/static/check-mark.png)
To calculate: The gradient vector and Hessian matrix for the function
Answer to Problem 5P
Solution:
The gradient vector for the function
The Hessian matrix for the function
Explanation of Solution
Given:
The function
Formula used:
The gradient vector for the function
The Hessian matrix for the function
Calculation:
Consider the function,
Partial differentiate the above function with respect to x,
Again, partial differentiate the above equation with x,
Partial differentiate the
Now, partial differentiate the function
Again, partial differentiate the above equation with y,
Partial differentiate the
Therefore, the gradient vector for the function is,
And, the Hessian matrix for the function is,
(b)
![Check Mark](/static/check-mark.png)
To calculate: The gradient vector and Hessian matrix for the function
Answer to Problem 5P
Solution:
The gradient vector for the function
The Hessian matrix for the function
Explanation of Solution
Given:
The function
Formula used:
The gradient vector for the function
The Hessian matrix for the function
Calculation:
Consider the function,
Partial differentiate the above function with respect to x,
Again, partial differentiate the above equation with x,
Partial differentiate the
Partial differentiate the
Now, partial differentiate the function
Again, partial differentiate the above equation with y,
Partial differentiate the
Partial differentiate the
Now, partial differentiate the function
Again, partial differentiate the above equation with z,
Partial differentiate the
Partial differentiate the
Therefore, the gradient vector for the function is,
And, the Hessian matrix for the function is,
(c)
![Check Mark](/static/check-mark.png)
To calculate: The gradient vector and Hessian matrix for the function
Answer to Problem 5P
Solution:
The gradient vector for the function
The Hessian matrix for the function
Explanation of Solution
Given:
The function
Formula used:
The gradient vector for the function
The Hessian matrix for the function
Calculation:
Consider the function,
Partial differentiate the above function with respect to x,
Again, partial differentiate the above equation with x,
Simplify furthermore,
Partial differentiate the
Simplify furthermore,
Now, partial differentiate the function
Again, partial differentiate the above equation with y,
Simplify furthermore,
Partial differentiate the
Simplify furthermore,
Therefore, the gradient vector for the function is,
And, the Hessian matrix for the function is,
Want to see more full solutions like this?
Chapter 14 Solutions
EBK NUMERICAL METHODS FOR ENGINEERS
Additional Engineering Textbook Solutions
Pathways To Math Literacy (looseleaf)
Precalculus
Elementary Statistics ( 3rd International Edition ) Isbn:9781260092561
Intermediate Algebra (13th Edition)
Probability And Statistical Inference (10th Edition)
APPLIED STAT.IN BUS.+ECONOMICS
- A garden hose attached with a nozzle is used to fill a 20-gal bucket. The inner diameter of the hose is 1 in and it reduces to 0.53 in at the nozzle exit. The average velocity in the hose is 8 ft/s and the density of water is 62.4 lbm/ft3. NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Determine the volume and mass flow rates of water through the hose. The volume flow rate of water through the hose is ft3/s. The mass flow rate of water through the hose is lbm/s. The change in time? What is the exit velocity?arrow_forwardA 23-ft3 rigid tank initially contains saturated refrigerant-134a vapor at 160 psia. As a result of heat transfer from the refrigerant, the pressure drops to 50 psia. NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Determine the final temperature. Use data from refrigerant tables. The final temperature is ºF.arrow_forwardA 23-ft3 rigid tank initially contains saturated refrigerant-134a vapor at 160 psia. As a result of heat transfer from the refrigerant, the pressure drops to 50 psia. NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Determine the heat transfer. The heat transfer is Btu.arrow_forward
- The shaft shown in the figure below is subjected to axial loads as illustrated. The diameters of segments AB, BC, and CD are 20mm, 25mm, and 15mm, respectively. If the modulus of elasticity of the material is 610 MPa. Determine the change of A to D lengtharrow_forwardDetermine the final pressure and temperature. The final pressure is kPa. The final temperature is ºC.arrow_forwardAir enters the 1-m2 inlet of an aircraft engine at 100 kPa and 20°C with a velocity of 184 m/s. Determine the volume flow rate, in m3/s, at the engine’s inlet and the mass flow rate, in kg/s, at the engine’s exit. The gas constant of air is R = 0.287 kPa·m3/kg·K. The volume flow rate at the engine’s inlet m3/s. The mass flow rate at the engine’s exit is kg/s.arrow_forward
- The ventilating fan of the bathroom of a building has a volume flow rate of 33 L/s and runs continuously. If the density of air inside is 1.20 kg/m3, determine the mass of air vented out in one day. The mass of air is kg.arrow_forwardA steady-flow compressor is used to compress helium from 15 psia and 70°F at the inlet to 200 psia and 600°F at the outlet. The outlet area and velocity are 0.01 ft2 and 100 ft/s, respectively, and the inlet velocity is 53 ft/s. Determine the mass flow rate and the inlet area. The gas constant of helium is R = 2.6809 psia·ft3/lbm·R. The mass flow rate is lbm/s. The inlet area is ft2.arrow_forward1. The maximum and minimum stresses as well as the shear stress seen subjected the piece in plane A-A. Assume it is a cylinder with a diameter of 12.7mm 2. Draw the Mohr circle for the stress state using software. 3. Selection of the material for the prosthesis, which must be analyzed from the point of safety and cost view.arrow_forward
- Elements Of ElectromagneticsMechanical EngineeringISBN:9780190698614Author:Sadiku, Matthew N. O.Publisher:Oxford University PressMechanics of Materials (10th Edition)Mechanical EngineeringISBN:9780134319650Author:Russell C. HibbelerPublisher:PEARSONThermodynamics: An Engineering ApproachMechanical EngineeringISBN:9781259822674Author:Yunus A. Cengel Dr., Michael A. BolesPublisher:McGraw-Hill Education
- Control Systems EngineeringMechanical EngineeringISBN:9781118170519Author:Norman S. NisePublisher:WILEYMechanics of Materials (MindTap Course List)Mechanical EngineeringISBN:9781337093347Author:Barry J. Goodno, James M. GerePublisher:Cengage LearningEngineering Mechanics: StaticsMechanical EngineeringISBN:9781118807330Author:James L. Meriam, L. G. Kraige, J. N. BoltonPublisher:WILEY
![Text book image](https://www.bartleby.com/isbn_cover_images/9780190698614/9780190698614_smallCoverImage.gif)
![Text book image](https://www.bartleby.com/isbn_cover_images/9780134319650/9780134319650_smallCoverImage.gif)
![Text book image](https://www.bartleby.com/isbn_cover_images/9781259822674/9781259822674_smallCoverImage.gif)
![Text book image](https://www.bartleby.com/isbn_cover_images/9781118170519/9781118170519_smallCoverImage.gif)
![Text book image](https://www.bartleby.com/isbn_cover_images/9781337093347/9781337093347_smallCoverImage.gif)
![Text book image](https://www.bartleby.com/isbn_cover_images/9781118807330/9781118807330_smallCoverImage.gif)