Find the solution u(x,y) of a Laplace equation in a rectangle 0 < x < a, 0 < y < b, the boundary and intial conditions are as follows: u(0,y) = 0,         u(a,y) = f(y),     for 0 < y < b u(x,0) = h(x),    u(x,b) = 0,         for 0 £ x £ a h(x) = (x/a)2 and f(y) = 1 – (y/b) Consider combining the solutions of two problems, one with homogeneous boundary conditions for u(a,y) = f(y) and the other with homogeneous boundary conditions for u(x,0) = h. (x)

Q: Find the directional derivative of f(x,y, z) = xy + yz + zx at the point P(1, 1, 1) in the direction…

A:

Q: if z=xarctan(y/z), find ∂z/∂x at x=0, y=1, z=1

A:

Q: Let q (y, z) = (2y + 13) * (29 − z). Calculate ∂q/∂y and ∂q/∂z

A:

Q: Isin , it (z, y) # (0, 0) 2. Determine if a function f : R → R defined by f (r, y) = 0, if (z, y) =…

A:

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 an equation for the tangent plane to x? + y – 8xy + 4x – 5y – 9z = 765 at the point (-9, 9,…

A:

Q: 1) For f(x, y,z) = x2 – (1 – y)? + z?. Find directional derivative of f at P(1, 1, 1) in the…

A: Given that  f(x,y,z)=x^2-(1-y)^2+z^2 Point P(1,1,1) Vector is a=(0,3,4)  To find the directional…

Q: The directional derivative of f(x, y) = x sin y at P.(1, n) in the direction of the vector 31-41

A: We have to find the divergence of the function first.

Q: F being a function of two variables;  F( x= az, y=bz) z = z(x, y) defined by the relation a∂z/∂x…

A: It is given that F is a function of two variables x and y and is given by Fx=az, y=bz. Here, z=zx,y.…

Q: 2. Let 3 – 4ry? 12 + y? if (x, y) # (0,0) f(x, y) = if (x, y) = (0,0) %3D (a) Is f differentiable at…

A:

Q: (a) (x² + y² + z² + w?)3/2 = 4

A:

Q: Traveling at a constant rate in a straight line, an object moves from the point (10, 3) to the point…

A:

Q: For f(x,y, z) = x² – (1 – y)² + z?. Find directional derivative of f at P(1,1,1) in the direction of…

A: The directional derivative of a function f  in the direction of the vector u is given by…

Q: Evaluate [(y²+z°)dx+(z²+x²)dy+(x²+y²)dz (0,0,0) to (1,1,1)

A: Here we need to evaluate ∫c(y2+z2)dx+(z2+x2)dy+(x2+y2)dz from (0,0,0) to (1,1,1).

Q: Using the information in the table below, find the occupancy rate for the years 2021 to 2024 using…

A: Given α=0.3. Let At represent the actual values of the occupancy rate and Ft represent the forecast…

Q: Find equations for (a) the tangent plane and (b) normal line at the point P on the given surface 1)…

A:

Q: |dx + N(x,y)dy = 0 x² + y,

A:

Q: Let r(t) = (cost,sin t, ln(cost)).Find T, N, and B at (1, 0, 0).

A: Let rt=cos t, sin t, lncos t at 1,0,0 Find T, N and B. Tangent vector to the curve is denoted by…

Q: Find the coordinate vector [x]B of the vector x relative to the given basis B. 23) b1 =| b2 and B =…

A:

Q: 1(x, y, z) = ln(2x + 4y + z²), at the point u = (−1,1,0), in the direction of the unit vector…

A:

Q: sin(xz – 1) In(6+ 3xz + 4yz) = 0 determines z as function of x and y in a neighbourhood of the point…

A: Claim: at (x,y)=(1,1) for the equation  Take partial derivative with respect to x Substitute…

Q: O Đ 11) Calculate the Jacobian, J, for the change of variables x = u cos(0) -v sin(e) and y = u…

A: The objective of this question is to calculate the Jacobian of the transformation from the variables…

Q: t f(x, y) = x² cos(xy). ) Determine f(x, y). Determine fy(x, y). Determine Vƒ(1, π).

A: Topic:- application of derivatives     Note:- as per our guideline if more than three subparts…

Q: One model for expenditures at high school gives the cosr C, in dollars per pupil, as a function of…

A: The quadratic equation, y=ax2+bx+c can be expressed in the vertex form as y=a(x-h)2+k, where, the…

Q: O Đ 11) Calculate the Jacobian, J, for the change of variables x = u cos(0) -v sin(e) and y = u…

A: The objective of this question is to calculate the Jacobian of the transformation from the variables…

Q: (B): use the directional derivative to estimate how much f(x, y) = cos(xy) + xy² Will change if the…

A:

Q: Find the Jacobian a(x, y) / a(u, v) for the indicated ch x = e-u sin v, y = e-U cos v a(x, y). a(u,…

A:

Q: Find the linearizations L(x, y, z) of the functions at the given points. ƒ(x, y, z) = tan-1 (xyz) at…

A: To find the linearization Lx,y,z of the function fx,y,z=tan-1xyz at 1,0,0