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.
Describe the two main geometric properties of the gradient V f.
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.
Use the gradient to find the directional derivative of the function at P in the direction of v.
g(a, y) = a2 + y² + 1, P(1,2), v (2, 3)
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University Calculus: Early Transcendentals (4th Edition)
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