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3250Jacobian of a Simple Nonlinear MapFor x=u 2-v,\ y=u+v 2, compute \partial(x,y) \partial(u,v) symbolically.数学中等derivation未尝试面试订阅3251Tangent Plane to a Quadratic SurfaceFind the tangent plane to z=x 2+y 2 at the point (1,2,5).数学中等derivation未尝试面试订阅3252Linearization of an Exponential SurfaceFind the linearization of f(x,y)=e x+2y at (0,0).数学中等derivation未尝试面试订阅3253Tangent Plane to a Product SurfaceFind the tangent plane to z=xy at (2,3,6).数学中等derivation未尝试面试订阅3254Linearization of a Logarithmic SurfaceFind the linearization of f(x,y)=\ln(x+y) at (1,1).数学中等derivation未尝试面试订阅3255Tangent Plane to a Three-Variable SurfaceFind the tangent plane to w=x 2+y 2+z 2 at (1,1,2,6).数学中等derivation未尝试面试订阅3256Classify the Origin for x^2+2y^2Classify the critical point at the origin for f(x,y)=x 2+2y 2.数学中等derivation未尝试面试订阅3257Classify the Origin for x^2-y^2Classify the critical point at the origin for f(x,y)=x 2-y 2.数学中等derivation未尝试面试订阅3258Classify the Origin for -x^2-4y^2Classify the critical point at the origin for f(x,y)=-x 2-4y 2.数学中等derivation未尝试面试订阅3259Classify the Origin for x^2+xy+y^2Classify the critical point at the origin for f(x,y)=x 2+xy+y 2.数学中等derivation未尝试面试订阅3260Classify the Origin for -x^2+2xy-y^2Classify the critical point at the origin for f(x,y)=-x 2+2xy-y 2.数学困难derivation未尝试面试订阅3261Area of an Ellipse from the Unit DiskThe map x=2u,\ y=3v sends the unit disk u 2+v 2\le 1 to an ellipse in the (x,y)-plane. What is the area of that ellipse?数学中等derivation未尝试面试订阅3262Area of a Parallelogram from a Unit SquareThe map x=u+v,\ y=u-v sends the unit square 0\le u,v\le 1 to a parallelogram. What is its area?数学中等derivation未尝试面试订阅3263Integral After a Simple ScalingUse the change of variables x=2u,\ y=3v to compute \iint R 1\,dx\,dy, where R is the image of the rectangle 0\le u\le 1,\ 0\le v\le 2.数学中等derivation未尝试面试订阅3264Area Scaling Under a Triangular MapThe map x=u,\ y=u+2v sends the unit square to a parallelogram. Compute the image area.数学中等derivation未尝试面试订阅3265Area of a Polar WedgeUse the polar Jacobian to compute the area of the region 0\le r\le 2,\ 0\le \le /3.数学中等derivation未尝试面试订阅3266Best Rank-One Reconstruction Error From Singular ValuesA centered data matrix has singular values \sigma 1\ge\sigma 2\ge\cdots\ge\sigma r. What is the Frobenius-norm reconstruction error of the best rank-one approximation?数学中等derivation未尝试面试订阅3268Recovering Correlation Strength from a Known First PCA covariance matrix has the form \Sigma=egin pmatrix 5&c\c&5\end pmatrix . You are told that the first principal component points along (1,1) and has variance 8. What is c?数学中等derivation未尝试面试订阅3269One-PC Reconstruction of a Two-Asset MoveLet \Sigma=egin pmatrix 9&-3\-3&9\end pmatrix , whose first principal direction is along (1,-1). For the observed move x=(2,-1), what are the rank-1 reconstruction using only the first principal component and the residual?数学中等derivation未尝试面试订阅3270How Standardization Changes the PCA SpectrumA raw covariance matrix is \Sigma=egin pmatrix 9&6\6&9\end pmatrix . After standardizing each coordinate to unit variance, what are the eigenvalues of the correlation matrix, and what fraction of standardized variance does the first component explain?数学中等derivation未尝试面试订阅