Let S be a surface described by x² + y² + 32² = 1, z ≤ 0, and let F(x, y, z) = yi-xj+zx³y²k. Prove that 1₁0x (V x F). dS = 2π

Q: Assume S is the surface x1000 + y8000 + z1000 = 1,000,000 and F = [9 + ee^(xyz), x8yz, 3x - 5y -…

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Q: Show that the curve r= Vti + Vtj + (2t – 1)k is tangent to the surface x +y² – z = 1 when t=1

A: Given that r=ti+tj+2t-1k and f=x2+y2-z-1 Take the derivative for f, ∇f=ddx,ddy,ddzf=2x,2y,-1…

Q: Find the equation of the tangent plane to the surface determined by x^3y^2+z−35=0 where x=3 and y=5.

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Q: Consider the function f(x, y) = (e - 3x) cos(y). Suppose S is the surface z = f(x, y). (a) Find a…

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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: Sketch the following surfaces. 1) 4x^2+9z^2=9y^2 2)y^2-x^2=z Show all work.

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Q: Find a non-zero vector B' which is perpendicular to A'= 2i'-3j'.

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Q: Suppose given a function f(x, y, 2) of three variables, and a point (o, y0, 20) €R. Explain in your…

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Q: If Φ = xy^2 + 6 z^3 x^2, find  ∇Φ and |∇Φ| at the point (6,2,2)​ ​

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Q: Find the tangent planes to the surfaces given by the equations z = 7x^2 - 12x - 5y^2 and xyz^2 = 2…

A: NOTE: Refresh your page if you can't see any equations. . the first surface is so we can write…

Q: ind the parametrization of the curve defined by the intersection of the two surfaces y = x^2 − 3z^2…

A: To find the parametrization of the curve defined by the intersection of the two surfaces  and ,…

Q: A surface has Cartesian equation :- f(xy). given by f(1.9)=6xy-5x -3y+ +7x-6y a) Use any…

A: To graph the function: z=f(x,y) where f(x,y)=6xy-5x2-3y4+7x-6y

Q: find    (a)    div(f)   (b)    curl(f) if f=x^2(vector i)+y^2(vector j+2yz(vector k)

A: Let's understand the mathematics behind the div and the curl functions. Please see the expressions…

Q: Determine the following: (a) the general equation of the tangent plane to the surface at Po (b) the…

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Q: Find an expression for a unit vector normal, no, to the surface x = (4 – cos (v)) cos (u), y = (4 –…

A: The complete solutions are given below

Q: An equation for the tangent plane to the surface (xty at the point P(1, 1, 4) is a) 8r + 8y-:= 0 b)…

A: The given surface is  f(x,y)=z=(x2+y2)2  ...........(1); we have to find the tangent plane at point…

Q: Find a vector that is perpendicular to the surface 4e^{(z-1)}+x^{2}y=xyz at the point (−1,−2,1).

A: Given the surface :

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

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Q: Find the unit normal to the surface z=2+2z^2-3y^3 at the point (1,-1,7), pointing down with respect…

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Q: If the circles x\power{2}+y\power{2}+2a'x+2b'y+c'=0 and 2x\power{2}+2y\power{2}+2ax+2ay+c=0…

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Q: y Consider the surface of equation z = X+Y² What is the equation for the tangent line to S at the…

A: This is a problem of vector calculus.

Q: Find a point on the surface z = 3x^2 − y^2 at which the tangent plane to this surface is parallel to…

A: We need to find the Point on the surface of  z=3x2−y2 given tangent plane to this surface is…

Q: Find the equation of the tangent plane to the given surface at the specified point   2x=y^2+z^2-9 at…

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Q: 3xyz = 63 and 3x² + 4y² + 8z² = 423 at point (−3, 1, −7). ř(t) = with -∞ < t <∞

A: The parametric form of a line that passes through x0, y0, z0 and has the direction ratios: a, b, c…

Q: Find the extreme values of the function f(x, y, z) = xy + z^2 on the circle in which the plane y - x…

A: Given function is fx,y,z=xy+z2. We have to find the extreme values of the given function on the…

Q: Find the parametrization of the curve defined by the intersection of the two surfaces y = x^2 − 3z^2…

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Q: consider the vector-valued function r(t) = a) State the domain of r(t) b) Identify where r(t) is…

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Q: Let S be the surface given by the equation x^3+y^3-xz=5. What is the equation of the tangent plane…

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Q: Find the equation of the tangent plane to the graph of f(x, y) = 8x² – 2xy? at the point (5, 4). A)…

A: We first find the partial derivatives  So the direction vector of the normal is = (48,-80)

Q: Find all points on the curve r(t) = ti + t^2 j + t^3k where its tangent line is parallel to the…

A: The curve is represented by the vector function r(t)=ti+t2j+t3k .To find the points on the curve…