Gradient fields Find the gradient field F = ▿ϕ for the potential function ϕ. Sketch a few level curves of ϕ and a few vectors of F . 25. φ ( x , y ) = x 2 + y 2 , for x 2 + y 2 ≤ 16
Gradient fields Find the gradient field F = ▿ϕ for the potential function ϕ. Sketch a few level curves of ϕ and a few vectors of F . 25. φ ( x , y ) = x 2 + y 2 , for x 2 + y 2 ≤ 16
Gradient fieldsFind the gradient fieldF = ▿ϕ for the potential function ϕ. Sketch a few level curves of ϕ and a few vectors ofF.
25.
φ
(
x
,
y
)
=
x
2
+
y
2
,
for
x
2
+
y
2
≤
16
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.
2. Write an inline function that returns the value of the function
.2
f(t, x) = sin(Va t) cos (Tx)
and also works for vectors. Test your function by plotting it over the region [0, 5] × [0, 5]. '
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.
φ(x, y) = y - 2x; A(-1, -2), B(0, 0), C(1, 2), and D(2, 4)
3. Let f(x, y) = sin x + sin y. (NOTE: You may use software for any part
of this problem.)
(a) Plot a contour map of f.
(b) Find the gradient Vf.
(c) Plot the gradient vector field Vf.
(d) Explain how the contour map and the gradient vector field are
related.
(e) Plot the flow lines of Vf.
(f) Explain how the flow lines and the vector field are related.
(g) Explain how the flow lines of Vf and the contour map are related.
Calculus, Single Variable: Early Transcendentals (3rd Edition)
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