Clairaut's Theorem Let DC R?be a disk containing the origin and assume that g: D R is a function given by g(x,y) = e" (cos y + asin y). Prove that g(,y) satisfies the Clairaut Theorem at point (0,0).

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A: Identify the graph of the vector function .From the equations it follows,,  and Now fix the value of…

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Q: Use Stokes' theorem to evaluate Ĝ - dĩ, vhere Ĝ(1, Y, 2) = (2r, y,0) and D is any closed curve.

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Q: Q. Let f(z)= u(xy)+ i V(x>Y) be analytic function and Z= re Find the Cauchy - Reimann eguations in…

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Q: a) By first sketching its projection onto the yz-plane, sketch the curve r(t) = (t, cos(t), 2…

A: For (a), We will plot the sketch of r(t)=t, cos(t), 2 sin(t),  0≤t≤4π onto the yz-plane (that means…

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A: Given the curve     rt=2sin(t),2cos(t),4sin2(t) and the point P1,3,1.

Q: 6. Let f(r, y, z) = e=² cosh y + z ar ff at the point P in the direction of :

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Q: Example Let f(r,y, z) = ryz sin rz. Find Vfo at the point Po = (1, –2, 7) and compute the…

A: to find : (1) grad f. (2) directional derivative.

Q: 7. If f is a scalar field and v = [v1, v2, v3] is a vector field, show that div(fv) = fdiv(v) +v •…

A: We will use simple definition of divergent of vector quantity and gradient of scalar quantity.

Q: TRUE OR FALSE. A plane curve C defined by the parametric equations x = f(t) and y = g(t), where t is…

A: check the given statement is true or not.

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Q: 7. For the vector-valued function √t-1 ln(2t - 1) t² - 1 r(t) = (VI- t-1 2 COS πt rt), (t (t = 1):…

A: Let us solve above problem

Q: Consider the function f(x,y) over the domain D enclosed by the 1+ x2 + y² polar curve r =2+…

A: (a) Look at the level curves of the function within the domain D.    The boundary of the domain D…

Q: Let be a curve defined by the vector function (t)= 6ti + çin 2t -cos 2+ k 3- and Find the moving…

A: Let C1 be a curve defined by the vector function R→(t)=6ti→+sin2tj→-cos2tk→. Find the moving…

Q: 2. Consider the curve traced out by the vector valued function r(t) = (sin t cos t, cos? t, sin t)…

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Q: Let y : IR → R be the parametric curve Y(t) = (, t, t). Find the coordinates of a point P on this…

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Q: 2. Find the length of the curve r(t) = e'i + e' sin 2tj + e' cos 2tk, 0st< 2n.

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Q: Use Green's theorem to evaluate F• dr. (Check the orientation of the curve before applying the…

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Q: Find T, N, and k for the space curve r(t) = (7e'cos t)i + (7e'sin t)j + 4k.

A: Given data:  r(t)=(7et cost) i +(7et sin t) j+4k

Q: 1. Find the tangent vector to the curve 6t r(t) = [t³ – t, t+1 , (2t + 1)*] at t = 1.

A: As per our guidelines we are supposed to answer only one question. If you want solution of any…