8. Choose, refer and use one of the theorems of Vector Calculus to evaluate the line integral F- ds where F = (6r2 + 3y2 - 2y?, -r? + 9ry?23 – 3y?, 9ry z2 – y? + 3) and C is the boundary of the surface with parametrization F(s, t) = (s, t, 4 – 3t) where 0 < s<1 and 0

Q: 32. Find the equation of a plane through the origin and per- pendicular to x – y+ z = 5 and 2x + y–…

A: Note- According to Bartleby guidelines, we are supposed to do only one question. If you want more to…

Q: An equation for the tangent plane to the surface : = (xty at the point P(1, 1, 4) is a) 8x + 8y -: =…

A: Equation of a tangent to the surface z=f(x,y) at point (a,b,f(a,b)) is given by…

Q: a) f : R? → R, f (x, y) = Vy sin x 5) f : R2 → R. f (r. u) = GGrul + In (a2 + 1? – 1). /4 – 72 -

A: We will find the domain

Q: 49. Let f(x, y, z)=sin(x² + y² +2²). (a) Describe in words the shape of the level surfaces of f. (b)…

A:

Q: Ollowing fünctión. T: R2 R2, T(x, y) = (x, 8) Find the following images for vectors u = (u,, u,) and…

A:

Q: Verify Green's Theorem in the plane for F = xyî + x²ĵ and where C is the boundary of he region…

A:

Q: Let C be the positively oriented square with vertices (0, 0), (2, 0), (2, 2), (0, 2). Use Green's…

A: Introduction: According to Green's Theorem, a line integral of a vector over a closed curve C…

Q: Consider the surface of equation z = x2 + y² . What is the equation for the tangent line to S at the…

A: We need to find equation of tangent line.

Q: Consider the function F(x, y, z)=(√√1 - x² - y², ln(e² - z2²)). (a) This function is a mapping from…

A:

Q: H. Use the gradient to find the equation of the tangent plane to each of the surfaces at the given…

A:

Q: 4. Consider the vector function r(z, y) = (x, y, x² + 2y*). (a) Re-write this vector function as…

A: For the given vector function, PART A The surface function is given by, f(x,y)=xi^+yj^+(x2+2y2)k^…

Q: Show that the function given by f(x,y)= x² + 3xy is differentiable at every poin using the…

A: Introduction: Distinctiveness (differentiability) implies continuity. If a function is…

Q: Is the set V = {(x, y)|x > 0} where (x1, Y1) O (x2, Y2) = (x1x2, Y1 + Y2) and c O (x, y) = (x°, cy)…

A:

Q: Let C be the positively oriented square with vertices (0, 0), (2,0), (2, 2), (0, 2). Use Green's…

A: Let the given square be OABC which is positively oriented.

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: find the points on the surface + z² = 1 that have their tangent plane perpendicular to the vector…

A: For tangent plane of the surface f(x,y,z)=k at a given point normal vector to tangent plane  will be…

Q: Consider the surface S of equation 2 = x² + y*" An equation for the tangent line tangent to S at…

A:

Q: Q1.

A:

Q: 10. Let D be the parallelogram spanned by the vectors (3, 4) and (5, 1). Apply the change of…

A:

Q: Valero's 5k Fun Run was held yesterday in the downtown district. The top runner 4 runners (one of…

A: Given 1. Ralph finished 1 minute after the competitor in the white shirt. 2. The competitor in the…

Q: C₁: r₁(s) C₂: r₂(t) C3 r3(u) = = = (s 2)i+ (2s² - 1)j + k 2t² i + (3t+7)j + (t+1) k (SER) (t = R)…

A: for the given vector functions for the curves C1,C2 and C3 To answer the given questions.

Q: 2. Determine the vector-function curve of intersection of the plane given by the equation x = 9/2…

A:

Q: Let C be the curve connecting (0,0,

A: Given that C be the curve connecting  (0, 0, 0) to (1, 4, 1) to (3, 6, 2) to (2, 2, 1) to (0, 0, 0).…

Q: Suppose that z is an implicit function of x and y in a neighborhood of the point P = (0, −3, 1) of…

A: Equation of a line passing through point (a, b, c) and parallel to vector &lt;p, q, r&gt; can be…

Q: Find: The shortest distance from the point (2, 1, 3) to the plane x+y+z= 1. = The shortest distance…

A: