Gradient fields on curves For the potential function φ and points A , B , C , and D on the level curve φ ( x, y ) = 0, complete the following steps. a. Find the gradient field F =∇ φ b. Evaluate F at the points A , B , C , and D . c. Plot the level curve φ ( x, y ) = 0 and the vectors F at the points A , B , C , and D . 44. φ ( x , y ) = 1 2 x 2 − y ; A (−2, 2), B (−1, 1/2), C (1, 1/2), and D (2, 2)
Gradient fields on curves For the potential function φ and points A , B , C , and D on the level curve φ ( x, y ) = 0, complete the following steps. a. Find the gradient field F =∇ φ b. Evaluate F at the points A , B , C , and D . c. Plot the level curve φ ( x, y ) = 0 and the vectors F at the points A , B , C , and D . 44. φ ( x , y ) = 1 2 x 2 − y ; A (−2, 2), B (−1, 1/2), C (1, 1/2), and D (2, 2)
Solution Summary: The author defines the gradient field F as the vector field which is obtained by the scalar-valued function, phi .
Gradient fields on curves For the potential function φ and points A, B, C, and D on the level curve φ (x, y) = 0, complete the following steps.
a. Find the gradient field F =∇φ
b. Evaluate F at the points A, B, C, and D.
c. Plot the level curve φ(x, y) = 0 and the vectors F at the points A, B, C, and D.
44.
φ
(
x
,
y
)
=
1
2
x
2
−
y
;
A(−2, 2), B(−1, 1/2), C(1, 1/2), and D(2, 2)
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
I would need help with a, b, and c as mention below.
(a) Find the gradient of f.(b) Evaluate the gradient at the point P.(c) Find the rate of change of f at P in the direction of the vector u.
Describe the two main geometric properties of the gradient V f.
Explain how a directional derivative is formed from the two partial derivatives f, and f,.
Choose the correct answer below.
O A. Form the dot product between the unit direction vector u and the gradient of the function.
O B. Form the dot product between the unit direction vectors.
O C. Form the sum between the unit direction vector u and the gradient of the function.
O D. Form the dot product between the unit coordinate vectors.
Click to select your answer.
Chapter 17 Solutions
Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)
University Calculus: Early Transcendentals (4th Edition)
Precalculus Enhanced with Graphing Utilities
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.