dA If A (u)is a differentiable vector function of u and||A (u) || = 1, prove that is perpendicular to A du

Q: Prove that if u is the vector function with the constant value C, then du/dt = 0.

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Q: 3. Let f(x, y) = x² – y². Find a unit vector i for which D7f(3, 4) Dif (3,4) is the directional…

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Q: Show that the vector function r defined by r(t) = ƒ(t)i + g(t)j + h(t)k is continuous at t = t0 if…

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Q: Let C be the curve of intersection of the paraboloid a² + y² = z and the cylinder +(y-1)² = 1 (a)…

A: Given that C is the curve of intersection of the paraboloid x2 + y2 = z and the cylinder x24 + y -…

Q: Let C be the curve of intersection of the paraboloid x² + y² = z and the cylinder + (y − 1)² = 1 (a)…

A: We will use the basic knowledge of vector calculus and differential calculus to answer all three…

Q: Find a vector-valued function whose graph is the indicated surface. The cylinder 9x² + y² = r(u, v)…

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Q: Consider the function  f(x,y,z) = sqrt(xyz)   what is the gradient of F?   Determine the…

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Q: Let f : R+ × R → R be the function f(x,y) = 2x + VTy³. (a) Find a tangent vector to the graph of…

A: The given function is: fx,y=2x+xy3 The given point is: 1,1,3

Q: Show that the graph of the vector-valued function r(t) = is a circle, and find its radius and…

A: The equation of a circle is defined by the equation, (x-h)2+(y-k)2+(z-l)2=r2 , where, (h,k,l) is the…

Q: Find the gradient vector field of f(x,y) = x2 + y2 and sketch it.

A: Solution

Q: Find the gradient vector field of f(x, y) ,3

A: Given,          The gradient vector field of fx,y=x6y3.

Q: if A=i+j+X, B=zi-j+3ksfind vector normal to B and in plane of A and B- ans. $2(3i+9j+x)

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Q: Find a vector function r(t) that represents the intersection of the surfaces: z = x² + y² z = y + 1

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Q: Prove that the tangent vector is always orthogonal tothe position vector for the vector-valued…

A: Given r(t) =⟨sin t, sint cos t, cos2 t⟩.  we need to prove that the tangent vector is always…

Q: If Ū is a unit vector and Di f(x, y) = 0 for all points (x, y) then f is a constant function..

A: We have to determine whether the following statement is true or false : If U→ is a unit vector  and…

Q: f f(x, y, z) = xy: on of vectorî - %3D

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Q: (b) Identify the unit vector u pointing in the direction of maximum increase of the function f(x, y)…

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Q: Determine if the vector filed is conservative and if so find the potential function F (x, y) = 12xyi…

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Q: If r(t) is a vector-valued function that is continuous on the interval [a, b), then r(t) dt is a…

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Q: Find a function f (x, y, z) such that ∇f is the constant vector〈1, 3, 1〉.

A: It is given that, ∇f=<1,3,1>   It is known that a vector ai^+bj^+ck^ can also be represented…

Q: Match the vector-valued function with its graph. r(u, v) = u cos(v)i + u sin(v)j + uk

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Q: Find a non-zero and non-constant vector function F(x, y) = F1(x, y)i+ F2(x, y)j, ƏF OF 02F and dy?…

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Q: Find the gradient vector of f(x, y) = e®"y at the point (1, In(2)). 6 In(2)i+ 2j O 2i +j 2i + 2j 3…

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Q: Find the gradient vector field of f. f(x, y, z) = 6/x?+ y² + z? Vf(x, y, z) =

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Q: use the parameter x=t to find a vector-valued function for the space curve represented by the…

A: Given surface x2 + z2=4 and x - y=0   and the parameter x = t

Q: (a) Find the domain of the vector function r(t) and sketch its range, where r(t) = (v25 – t², 1, vf²…

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Q: -dt. dF = (dr · V)F+ at

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Q: Find the directional deriu atrue Du f(-3,-1,3) for the function f (xiy,z) =xy + yz +Z X in the…

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Q: Differentiate the vector-valued function r(t)=⟨sin(t2),4t2−t⟩.

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Q: Which of the following is NOT true about the vector function r = r(t) where |r |= constant

A: yes, given statement is true. for a given vector function r = r(t)r = constant always true

Q: Why is r(t) = (f(t),g(t).h(t)) called a vector-valued function? %3D

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Q: Find a vector function, r(t), that represents the curve of intersection of the The cylinder x² + y2…

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Q: Give the two unit vectors u such that the function z = g(x, y) of example 3 has zero derivatives at…

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Q: Why is r(t) = ⟨ƒ(t), g(t), h(t)⟩ called a vector-valued function?

A: Given r(t) = ⟨ƒ(t), g(t), h(t)⟩. We need to give reason why r(t) is called vector-valued function.

Q: Describe a non-zero vector function r(t) in R^3 whose acceleration vector a(t) satisfies a(t) =…

A: Solution is given below:

Q: Suppose r(t) is a vector function in R3 Show that if r(t) has a constant speed, then r′(t) is…

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Q: Sketch some vectors in the vector field F(x, y) = 2xi + yj.

A: Given: F(x, y) = 2xi^ + yj^

Q: Let f(x,y)=y2 +sin(y+T)+x +cos(xe). At the point (TT,0) compute the unit vector in the direction of…

A: We have to find the unit vector in the direction of the maximum increase of the function f. This…

Q: a) Find the directional derivative f (x, y, z) = x In(y+z) at point P (2, e, e) in the direction of…

A: We have to find directional derivative of vector f in the direction of vector v:

Q: (b) Identify the unit vector u pointing in the direction of maximum increase of the function f(x, y)…

A: Find your answer below

Q: Usé the parameter x = t to find a vector-valued function for the space curve represented by the…

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Q: Find the gradient vector field of f. f(x, y) = xe³xy Vf(x, y) =

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Q: Find the directional derivative of the function at the given point in the direction of the vector v.…

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Q: 1) Find a vector function r(t) that represents the intersection of the surfaces: = /x² + y² z = y +…

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Q: Suppose a vector function F(x, y) = 3xi + 2xyj + y ln zk moves along a path C where x = t2, y = t +…

A: The formula to find the work done by the vector function helps to find the required work done. The…

Q: Find a vector-valued function whose graph is the indicated surface: the part of the paraboloid z =…

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Q: B/ show that if u(t) = d cos t + b sin t, where a and b are constant vectors |du d'u dt dt then…

A: B. Given: u(t)→ =a→ cos t+b→ sin t  where a→  and b→du⇀dt=-sin t a→ + cos t  b→d2u⇀dt2=-cos t a→ -…

Q: Let f(x, y, z) = x+y+z. What is the directional derivative of f in the direction of the unit vector…

A: The function is given by fx,y,z = x+y+z Unit vector is given by u = a,b,c To evaluate : The…