Let G be a solid with the surface
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- a) Find the directional derivative of f(x, y, z) = x²y²z + 4xz at (-1,2,1) in the direction 2i+j+ 2k.arrow_forward5. Find the directional derivative of the scalar function of (x, y, z) = xyz in the direction of the outer 27 normal to the surface z = xy at the point (3, 1, 3). Ans.arrow_forwardUse Green's Theorem in the form of this equation to prove Green's first identity, where D and C satisfy the hypothesis of Green's Theorem and the appropriate partial derivatives of f and g exist and are continuous. (The quantity ∇g · n = Dng occurs in the line integral. This is the directional derivative in the direction of the normal vector n and is called the normal derivative of g.)arrow_forward
- a) Compute the gradient of the function f(x, y, z) = e*y² . b)Compute the directional derivative Daf (1, 2, 3), 6. where 1 1 /2' 2' 2 c)Let p(t) be a parametric curve with P(7) = (1, 2, 3), (7) = (-2,5, 3). Compute the derivative (fop)'(7).arrow_forwardfind the directional derivative of f at p in the direction of B. if f(x,y,z)=x+ycos(z) where p denote the point (1,2,0) and B =(2,1,-2)?arrow_forwardSuppose the solid W in the figure is the spherical half-shell consisting of the points above the xy-plane that are between concentric spheres centered at the origin of radii 4 cm and 10 cm. Suppose the density 8 of the material increases linearly with the distance from the origin, and that at the inner surface the density is 8 g/cm³ while at the outer surface it is 10 g/cm. (a) Using spherical coordinates, write d as a function of p. Enter p as rho. 8(e) = 25/9(rho-4) (5.0 (b) Set up the integral to calculate the mass of the shell in the form below. If necessary, enter o as phi, and 0 as theta. B D CLI 25/9(rho-4)rho^2sinphi "OP Ópdp A = 0 B = 2pi C = 0 D= pi/2 (Drag to rotate) E- 4 F= 10 (c) Find the mass of the shell. 4536piarrow_forward
- Suppose that r is the position vector of a particle moving along a plane curve and dA / dt is the rate at which the vector sweeps out area. Without introducing coordinates, and assuming the neces- sary derivatives exist, give a geometric argument based on incre- ments and limits for the validity of the equation dA r X i . dtarrow_forwardConsider the function f : R³ → R defined by f(x, y, z) = tanh(5x² + 2y² - 3z²) and the point P: (x, y, z) = (1, 1, 1). (a) In what direction is f most rapidly increasing at the point P? (b) Calculate the directional derivative of f in the direction of u = (2, -1, 2) at the point P. (c) Determine an expression for the linear approximation f of f at point P. (d) Use the linear approximation to approximate the value of f(-).arrow_forwardFind an expression for a unit vector normal, no, to the surface x = (4 – cos (v)) cos (u), y = (4 – cos (v)) sin (u), z = sin (v) at the image of a point (u, v) for –x < v < a and -a < u < a. (Write your solution using the form (*,*,*). Use symbolic notation and fractions where needed.) no = Identify this surface. ellipsoid hyperboloid torus cylinderarrow_forward
- 3. Let F and G be vector ficlds with differentiable components. Express curl (F x G) in term of div and dot products.arrow_forwardH. Use the gradient to find the equation of the tangent plane to each of the surfaces at the given point. a) x² + 3x²y-z = 0 at (1,1,4) (Answ: 9x+3y-z = 8) b) z = f(x, y, z) = r²y³z at (2,1,3) (Answ: 4x - 3y -z = 2) I. In electrostatics the force (F) of attraction between two particles of opposite charge is given by (Coulomb's law) where k is a constant and r = (x, y, z). Show that F is the gradient T (Hint: ||||||(x, y, z)||). Important problem! F(r) = k₁ of P(7) ||7-1³ -k ||1| =arrow_forwardFor an implicit surface determined by the implicit relation function F (x, y, z) =0, it is known that its tangent plane is given by 3x −y + 2z = 12 at the pointP (5, 5, 1). Find the directional derivative ∇uF (P ) where u = (2, 2, −1)arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage