Laplace equation Uxx + Uyy = 0 under the boundary conditions: U(0,y) = U(x,0) = 0, Ux(1,y) = 0, and U(x,1) = f(x).

Ex.: Find the directional derivative of the function f(x, y) = x'y* + x*y* at the point (1,1) in the direction of the vector (4,3).

1. (+y + ) (1+? + +) 9. Integrate the function f(r,y, 2) = over the entire 3D coordinate system, i.e., all of R'.

1) Let f(x, y) = -4x + 3y. Find the directional derivative in the direction of v = <3, 1> from the point P = (5, 2).

a) Find the directional derivative of f(x, y, z) = x²yz + 4xz² at (1, -2, 1) in the direction 2i - j - 2k.

b) Find the directional derivative of f (x,y) = xy + e*°vy + 1 at (-1,1) in the direction of the origin.

Suppose F(x, y) = (5x – y)i + x}. (a) Find a vector parametric equation for the line segment from the origin to the point (4, 16) using t as a parameter. F(1t) = (b) Find the line integral of F along the line segment from the origin to (4, 16). (c) Find a vector parametric equation for the parabola y = x from the origin to the point (4, 16) using t as a parameter. F(t) = (d) Find the line integral of F along the parabola y = x² from the origin to (4, 16). (e) True or False: The line integral of any vector field F from point P to point Q is the same no matter what path is taken from P to Q. ?

(3) Consider the derivative dz 2xydx + (x2 + 2y)dy Evaluate whether this is a total derivative, by integrating along straight lines (i) (x1, y1) → (x2, Y1) then (x2, y1) → (X2, Y2); and (ii) (x1, Y1) → (X1, Y2) then (x1, y2) → (x2, y2).

Evaluate |f(x,y) ds where Cis the portion of the curve y = In x from the point (1,0) to the point (e,1) and C ху f(x, y)=- Vx? +1