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hwk4 - From the textbook solve 1 9 1 and 1 9 2 6 Friday...

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Math 31BH - Homework 4. Due Tuesday, February 8. 1. ( Wednesday, January 26. ) Chain rule. From the textbook, solve 1 . 8 . 2 , 1 . 8 . 10( a ) , 1 . 8 . 11 . 2. ( Wednesday, February 3. ) Euler’s identity. A function f : R n R is said to be homogeneous of degree d if f ( tx 1 , . . . , tx n ) = t d f ( x 1 , . . . , x n ) for all real numbers t, x 1 , . . . , x n . (i) Show that f ( x 1 , . . . , x n ) = x d 1 + . . . + x d n is homogeneous of degree d . (ii) Show that if f is homogeneous of degree d , then x 1 · ∂f ∂x 1 + . . . + x n · ∂f ∂x n = df. Hint: Differentiate the defining equality with respect to t and then set t = 1 . 3. ( Wednesday, February 3. ) Laplacian in polar coordinates. Using the chain rule, express the Laplacian of a function f in polar coordinates Δ f = f rr + 1 r f r + 1 r 2 f θθ . 4. ( Wednesday, February 3. ) The Laplacian, harmonic functions and orthogonal matrices. Let f : R 2 R be a harmonic function and let A be an orthogonal 2 × 2 matrix. Show that g ( x ) = f ( Ax ) is a harmonic function. 5. ( Friday, February 5. ) Pathological functions.
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Unformatted text preview: From the textbook, solve 1 . 9 . 1 and 1 . 9 . 2 . 6. ( Friday, February 5. ) Pathological functions and second order derivatives. Put f ( x,y ) = ( xy · x 2-y 2 x 2 + y 2 if ( x,y ) 6 = (0 , 0) if ( x,y ) = (0 , 0) . (i) Calculate the first derivative f x at (0 , 0) directly from the definition. (ii) Calculate the first derivative f x at any other point ( x,y ) 6 = (0 , 0) . (iii) Differentiate once more i.e. calculate f xy (0 , 0) using the definition. Confirm that f xy (0 , 0) = 1 . (iv) Repeat for the derivative f y (0 , 0) then f yx (0 , 0) . Confirm that f yx (0 , 0) =-1 . Observe that f xy (0 , 0) 6 = f yx (0 , 0) . 7. ( Friday, February 5. ) Spaces of matrices. From the textbook, solve 1 . 8 . 13 and 1 . 10 . 30 . 1...
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