The vector (x, 2y, –3z) is normal to the surface x2 + 2y? – 32 – 4 = 0 at any point (x, y, z). - - -

Q: Find the plane tangent to the surface z=x x+ y° at the point (3,4,25). a. 6x+8y-z=25 O b. 6x-8y-z=25…

A: As we know that; The equation of the tangent plane: nxx-x0+nyy-y0+nzz-z0=0 And The normal vector at…

Q: z dv, where E is enclosed by the paraboloid z = x2 + y² and the plane z = 4. Evaluate

A: see 2nd step

Q: Find an equation of the plane tangent to the surface xy z = 2 at the point (2, 1, -1). O A.x + 6y +…

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Q: (a) Prove that M: (x² + y^y° + 3z² = 1 is a surface. (b) For which values of e is M: z(z – 2) + xy =…

A: Consider the provided question, (a)

Q: Use Stoke’s Theorem to evaluate the line integral ∮C(y+2z)dx +(x+2z)dy +(x+2y)dz, where C is the…

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Q: A point moves along the intersection of the elliptic paraboloid z = 9x? + 3y² and the plane.x…

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Q: Find an equation for the tangent plane to the surface z = 6x3y2 + 3y at the point P(1, -3). Hint:…

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Q: Evaluate ſ F . n dS if F = (x + y² ™)ī − (2x)j + (2yz)k and $ is the surface of the plane 2x + y +…

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Q: B Suppose we want to find all the points on the surface gx, y) x+ 4y - 4 where the tangent plane is…

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Q: a) Evaluate x²y? ds, where the curve C is the intersection between the surface z = := V1– y? and the…

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Q: Find the flux of F (x, y, z) = 4.xi+ 4yj+ 2k outward (away from the z-axis) through the surface cut…

A: The field lines of a vector field F through surfaces with unit normal n, the angle from n to F is θ.…

Q: F = (x² – y²,x, 2yz); C is the boundary of the plane z = 6 – 2x – y in the first octant.

A: Given, F=x2-y2,x,2yz Find the line integral ∮CF·dr in the following. Here C is the boundary of the…

Q: Find the equation of the tangent plane to *+ y? at the point (2, –3, 10) the surface z = 4 O x + 4y…

A: We have to find

Q: If (a,B.y) is a point at which the surface x2+y2-z²-2x+12=0 has a horizontal tangent plane, then y|…

A: As per guidelines, we will solve first question only. Let α,β,γ be the point on the surface…

Q: a) Evaluate (-ri-yj+2z k) -n dS, where o is the portion of the paraboloid z = 3r + 3y? between the…

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Q: Sketch the surface z = 3x? + y? +1 and find its linear approximation at P-(0, -1)

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Q: Consider x = h(y, z) as a parametrized surface in the natural way. Write the equation of the tangent…

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Q: a) Find the points at which the twisted cubic F(t) = ti + t} +tk intersects the plane 4.x +2y+z = 24…

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Q: Find a vector normal to the surface x2 + y2 - z² = 18 at P = (3, 5, 4). %3D

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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: 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: d the normal vector to the surface y = 16x at (4,8). (a) 16(f-j) (b) (1 – f) (c) 6(1 – 1) (d) none…

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Q: a) Evaluate [-ri-yj+2: k) - n ds, (-a i- where a is the portion of the paraboloid z = 3r + 3y…

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Q: Find an equation of the plane tangent to the following surface at the given points. z= eW; (4,0,1)…

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Q: Find the equation of the plane perpendicular to the curve r(t) = (cos(2t), t, sin(2t)) %3D at the…

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Q: a) Evaluate y° ds, where the curve C is the intersection between the surface z and the plane x = 3…

A: Integration problem...

Q: What is the equation of the plane to the surface z = e3Y sin(3x) +2 at the point P,0, 3) O x- y + ez…

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Q: Find all points on the surface xy - z2 = 1 which are closest to that origin. Let x,y,z be the…

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Q: Let F(r, y, z) = 2x² + 14y? + 722. Find the equation to the tangent plane to the level of surface…

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Q: At what point on the curve x = t³, y = 6t, z = t4 is the normal plane parallel to the plane 6x + 12y…

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Q: Calculate the positive value of m so that the x+y+2z=m plane is tangent to the spherical surface…

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Q: Q3/ Sketch the surface z 3x2 + y? +1 and find its linear approximation at P.(0,-1).

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Q: Find a vector normal to the surface x2 + yz = 5 at (2, 1, 1).

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Q: The unit normal vector to the surface x'y + 2xz? = 8 at the point %3D (1,0, 2) is

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Q: Calculate F- dS if F(x, y, z) = x²ye²i+ æy?j – 2xye*k, where S is the surface of the box bounded by…

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Q: Show that the surfaces x2 + 2y2 + 3z2 = 3,   x2 + y2 + z2 + 6x − 10y + 14 = 0, are tangent to each…

A: Concept:   The calculus helps in understanding the changes between values that are related by a…

Q: a) Evaluate Fự ds, where the curve C is the intersection between the surface z = V1- y? and the…

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Q: : The curve C is intersecting by 12r² +9y? = 36 cylinder and - V3r plane from the point (0, 2,0) to…

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Q: a) Evaluate ds, where the curve C is the intersection between the surface z = V1- y? and the plane r…

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Q: e the electric flux through a paraboloid z = x² + y² – 16 that lies . It is oriented by an outward…

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Q: find the flux of f=zk across the portion of the sphere x2+y2+z2=a2 in the first octant in the…

A: Given the field, F=zk The surface S is the portion of the sphere x2+y2+z2=a2 in the first octant in…

Q: Find an equation for the tangent plane to the surface z = 3y^2 − 2x^2 + x at P(2, −1, −3). Express…

A: Let's find equation of tangent plane.

Q: Find an equation of the surface of revolution by revolving the plane curve 3y = v(9- [(x-1)] ^2)…

A: Solution

Q: Find the plane tangent to the surface z=x+ y at the point (3,4,25). a. 6x+8y-z=25 b. 6x-8y-z=25 c.…

A: We have to find the plane tangent for the given surface

Q: What is the perpendicular vector to the surface f(x.y.a)-ay-3r+'y-0 at (1,-1,2)? (3,3.0) (1,1.3)…

A: A vector that makes an angle of 90 degrees with the given surface is said to be perpendicular to the…

Q: A vector normal to the surface y-:-3 at the point (-1,-2, 2) is (a) i+2j+k (b)-i+j+2k) (e) (21 -j+k)…

A: The given surface is: xy-z=3 And, the given point is: -1, -2, 2 Since, the gradient represents a…

Q: yz ds where o is the part of the plane x + y+ z = 1 that lies in the first octant.

A: Given The given integral ∫∫σyzds over the region x+y+z=1 that lies in the first octant can evaluated…

Q: Use Stoke’s Theorem to evaluate the line integral ∮C(z2−y2)dx +(x2−z2)dy +(y2−x2)dz, where the curve…

A: Given equation of the curve is ∮C(z2-y2)dx+x2-z2dy+(y2-x2)dz ,the equation of the paraboloid…