Show that the given curve c(t) is a flow line of the given velocity vector field F(x, y, z). c(t) = (sin(t), cos(t), 3e¹); F(x, y, z) = (y, -x, z) (cos(t), - sin(t), 3e¹) c'(t) = F(c(t)) = Since c'(t) X F(c(t)), we have shown that c'(t) is a flow line of F.

Q: For an equation " + a? = 0 a) Draw a tangent vector to a solution (2(t), x'(t)) in the phase plane…

A: (a) To solve the given differential equation: Let   p=dxdt, or, x' ⇒dpdt=d2xdt2 ... differentiating…

Q: 5. Find the directional derivative of f(x, y) = xy – y at the point (1, 2) along the unit vector…

A: The given function is: f(x,y)=xy-y2 Given point is: P = (1, 2) Direction vector is: v=3, 4

Q: 4. (a) Determine the directional derivative of f(x, y, z) = zety at the point (1, −1, 1) in the…

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Q: Find the vector equation of the tangent line to the curve given by 7(t) = sinti + cos (2t)j + tan tk…

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Q: Find the derivative of the vector function r(t) = ln(19 - t²)i + √17+tj − 7e-¹t k - r' (t) = (¯¯

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Q: Find the directional derivative of f(x, y) = exy in the direction u = (,) at (xo, yo): (0,1). 3545…

A: Given query is to find direction derivative.

Q: Suppose f(x, y) = sin( (a) Vf(x, y) = = (b) Vƒ(3, π) = 2y x T 2y 2y 2y cos cos 2) Con (2) 200m (20 -…

A: Given that , fx,y=sin2yx and u is the unit vector in the direction of 1,1.

Q: Let 7 (t) = (t + 5, 5e“, 3 sin(5t)) Find the unit tangent vector T (t) at the point t = 0 T(0) = <

A:

Q: 2. Let T(x; y) = sin(2x – y) , P (-; ) and Q(0;0) (a) Determine the directional derivative of T at P…

A: Tx,y=sin2x-y, P=-π3,π6,𝐐=0,0                                      P1  P2                𝐐1,…

Q: f (x, y) is a differentiable function with Vf(4,3) = 7i+ 5j. 2. Suppose w = (a) Find the directional…

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Q: You are given the derivative of a vector function r in the component form is dr (e,3e",-2t). You are…

A: Given that: drdt=&lt;e-t, 3e3t, -2t&gt;          and r(0)=2i-j+k

Q: Find the directional derivative of f(x, y, z) = ((x+y+z)e², (x² + y²)e-²) at c = (1, 0, 1) in the…

A: The given problem is to find the directional derivative of the given vector function at point…

Q: Let w = : F(x, y, z)=x² tan(yz) + e³+1 ln(xy) and let P be the point (1, e, 0). (a) Find the rate of…

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Q: Consider the function f(r, y) = e=² sin( q² + y²) (a) ) Compute the gradient of f. (b) 1 ) Find the…

A: Given fx,y=e-xsinx2+y2

Q: II. Let F and G be vector-valued F(1) = (1,0,2), F'(1) = (1,1,2), G(1) = (2,1,3), G'(1) = (-1,…

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Q: Find the derivative of the vector function r' (t) = ( -> r(t) = ln(5t²)i + √√19+tj-8e²t k ID 9 9

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Q: Find the derivative of the vector function r(t) = In(10 – t) i+ v10+tj+ 8e-8* k r'(t) = (

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Q: Find a tangent vector of the curve7(t) = (t², 2 sin(t), 2 cos(t)) at (0,0, 2) (0,0,1) (1,0,0) None…

A: Here, Find it’s derivative This is the tangent at any point on the curve.

Q: a) Compute the gradient of the function f(x, y, z) = e*y² . b)Compute the directional derivative Daf…

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Q: Let w = F(x, y, z) = -x² tan(yz) + e²+1 In(ry) and let P be the point (1, e, 0). (a) Find the rate…

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