Problem Solving Ideas: Problem Description: There is a Gaussian surface S in three-dimensional space, and its parametric equation is: Mathbf(U,V)=(UCOS(V),usin(V),E where (0leQuleq1),(0leqVleq2Pi), and the surface components of the multivariate function (f(x,y,z)=xzy 2) are required to be calculated. Solution: Step 1: Determine the integral variables and rangesAccording to the parametric equation of the surface, we need to convert the multivariate function f into a parametric form. i.e., f(mathbf(u,v)))=ucos(v)e (usin(v)) 2 Step 2: Write the surface division expression Using the basic formula of the surface integral, we get: iintsf(x,y,z)ds=iintdf(mathbf(u,v)))|mathbfutimesmathbfv||dudv(mathbfu) and (mathbfv) represent the tangent vectors of the surface in the you and v directions, respectively, they are: mathbfu=(cos(v),sin(v),-2ue )mathbfv=(-usin(v),ucos(v),0)Step 3: Calculate the cross product of the tangent vector and its modulusCalculate the cross product of these two vectors: mathbfutimesmathbfv=(2ue ucos(v), 2ue usin(v),u 2) has a modulus length of: ||mathbfutimesmathbfv||=sqrtucos(v)) 2(2ue usin(v)) 2(u 2) 2}=ue sqrtu 2}Step 4: Calculate the surface division and substitute the above results into the surface integral expression to obtain the integral that needs to be computed at the end: iintdue sqrtu 2}(cos(v)e u 2sin 2(v))dudvStep 5: Actually calculate the integralSince this integral involves multiple variables and is nonlinear, the solution in closed form cannot be given here, It needs to be solved with the help of numerical integration methods. Methods such as Simpson's rule, trapezoidal rule, or Gaussian integrals can be used to integrate you and v respectively at the given intervals 0,1 and 0,2 to obtain a more final result. The above is the general idea and process to solve this problem, and the actual operation also needs to use mathematical software or programming language to carry out specific calculations of numerical integration. 