24. Outward flux of a gradient field Let S be the surface of the portion of the solid sphere x² + y² + z? < a² that lies in the first octant and let f(x, y, z) = lnVx² + y² + z?. Calculate Vf•n do. (Vf•n is the derivative of f in the direction of outward normal n.)

Q: 4. Find the flux of F across S if F = yi-xj+2zk and S is the hemisphere x² + y² + z² = 4, z ≥ 0,…

A: We have to find flux passing through sphere.

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: 3. (a) Show that the two surfaces S1 : z = xy and S2 : z =3x^2/4 - y^2 perpendicularly at the point…

A: As per our guideline, we are supposed to solve only first subpart. Kindly repost other subpart as…

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: 3. Find the directional derivative of the given function f at the given point P in the direction of…

A: 3. From the given problem:   fx,y,z=x2+y2+z2 ; p1,2,2 ; a=3,2,1   As we know that the expression for…

Q: Find div(curl F) = V. (V x F). F(x, y, z)= xyzi + yj + zk

A: If vector field then from the question

Q: Let z = g(x, y) = f(3 cos(xy), y + e™v) provided that f(3, 5) = 5, f1(3, 5) = 2, f2(3, 5) = 5. i)…

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Q: Find the tangent plane of the surface z = f(x, y) at (0,0, 0) in which 3y f(2, y) = || cos /3 t dt.

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Q: Let 2r + 3y –= 5 be the tangent plane to a surface z = f(r,y) at the point (1,1). (a) Find the…

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Q: The velocity field for a fluid is given by V=(X2t3)i+(2XYT)j. The convective acceleration in the…

A: Given, V=x2t3i^+2xytj^ Here, u=x2t3 and v=2xyt Convective acceleration in y-direction is given by:…

Q: Let f(x, y, z) = x cos z + y sin z and P = (−1, 1, 4). (a) (b) (c) at P. | Find the rate of change…

A: Here given function as fx, y, z=xcosz+ysinz

Q: F(1)=r cot"(1.4-r')i +1.4(r² -1) j + Jar +1k t = 0 at and is normal to the plane x+y-z 3

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Q: 10 12 14 10 12 14 (b) Calculate dB at t = 2. dt 64 dollars per month (c) Explain what the number you…

A:

Q: 1. Let D = the sales of DVDs in billions of dollars, S = the total sales of streaming videos in…

A: The sales of dvds given as the function D = -0.92t + 8.75 and the total sales of streaming videos…

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

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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: Find equations for (a) the tangent plane and (b) normal line at the point P, on the given surface x*…

A: According to bartleby guidelines when student asks more than one question expert have to answer…

Q: Q4) Consider the function f(x,y,z) = x³ – xy² – zy a) Find the derivative of f(x, y, z) in the…

A: Given function fx,y,z=x3-xy2-zy a) Directional derivative, D=∇f·A⏜…

Q: The plane y 1 intersects the surface z = x' + 5xy – y in a certain curve. Find the slope of the…

A: To find the slope of the tangent line of the curve: z=x7+5xy-y4 And the plane y=1, at the point…

Q: Find the tangent plane to the surface f(x, y) = e" cos y +e*sin y at (-1, ;). %3D OAz=-e(z+…

A: Option A is correct option.

Q: y Consider the surface of equation z = What is the equation for the tangent line to S at x2+y4 * the…

A: Solution

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

A:

Q: In spherical coordinates, (r, 0, ø), the gradient takes the form 1df 1 ðf Vf (r, 0, ø) : -î + -0 +…

A: