Lim 380 HW 2

108 days ago by ekalim

x, y= var('x,y') 
       
f(x,y)= (x)/(x^2+y^2) 
       
f.diff(x) 
        
(x, y) |--> -2*x^2/(x^2 + y^2)^2 + 1/(x^2 + y^2)
(x, y) |--> -2*x^2/(x^2 + y^2)^2 + 1/(x^2 + y^2)
show (f) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left( x, y \right) \ {\mapsto} \ \frac{x}{x^{2} + y^{2}}
\newcommand{\Bold}[1]{\mathbf{#1}}\left( x, y \right) \ {\mapsto} \ \frac{x}{x^{2} + y^{2}}
show (f.diff(x)) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left( x, y \right) \ {\mapsto} \ -\frac{2 \, x^{2}}{{\left(x^{2} + y^{2}\right)}^{2}} + \frac{1}{x^{2} + y^{2}}
\newcommand{\Bold}[1]{\mathbf{#1}}\left( x, y \right) \ {\mapsto} \ -\frac{2 \, x^{2}}{{\left(x^{2} + y^{2}\right)}^{2}} + \frac{1}{x^{2} + y^{2}}
show (f) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left( x, y \right) \ {\mapsto} \ \frac{x}{x^{2} + y^{2}}
\newcommand{\Bold}[1]{\mathbf{#1}}\left( x, y \right) \ {\mapsto} \ \frac{x}{x^{2} + y^{2}}
f.diff(x,x) 
        
(x, y) |--> 8*x^3/(x^2 + y^2)^3 - 6*x/(x^2 + y^2)^2
(x, y) |--> 8*x^3/(x^2 + y^2)^3 - 6*x/(x^2 + y^2)^2
f.diff(y,y) 
        
(x, y) |--> 8*x*y^2/(x^2 + y^2)^3 - 2*x/(x^2 + y^2)^2
(x, y) |--> 8*x*y^2/(x^2 + y^2)^3 - 2*x/(x^2 + y^2)^2
f.diff(x,x)+f.diff(y,y) 
        
(x, y) |--> 8*x^3/(x^2 + y^2)^3 + 8*x*y^2/(x^2 + y^2)^3 - 8*x/(x^2 +
y^2)^2
(x, y) |--> 8*x^3/(x^2 + y^2)^3 + 8*x*y^2/(x^2 + y^2)^3 - 8*x/(x^2 + y^2)^2
show (f.diff(x,x)+f.diff(y,y)) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left( x, y \right) \ {\mapsto} \ \frac{8 \, x^{3}}{{\left(x^{2} + y^{2}\right)}^{3}} + \frac{8 \, x y^{2}}{{\left(x^{2} + y^{2}\right)}^{3}} - \frac{8 \, x}{{\left(x^{2} + y^{2}\right)}^{2}}
\newcommand{\Bold}[1]{\mathbf{#1}}\left( x, y \right) \ {\mapsto} \ \frac{8 \, x^{3}}{{\left(x^{2} + y^{2}\right)}^{3}} + \frac{8 \, x y^{2}}{{\left(x^{2} + y^{2}\right)}^{3}} - \frac{8 \, x}{{\left(x^{2} + y^{2}\right)}^{2}}
f(x,y)=(x^2)/(x^2+y^2) 
       
show (f) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left( x, y \right) \ {\mapsto} \ \frac{x^{2}}{x^{2} + y^{2}}
\newcommand{\Bold}[1]{\mathbf{#1}}\left( x, y \right) \ {\mapsto} \ \frac{x^{2}}{x^{2} + y^{2}}
f.diff(x,x)+f.diff(y,y) 
        
(x, y) |--> 8*x^4/(x^2 + y^2)^3 + 8*x^2*y^2/(x^2 + y^2)^3 -
12*x^2/(x^2 + y^2)^2 + 2/(x^2 + y^2)
(x, y) |--> 8*x^4/(x^2 + y^2)^3 + 8*x^2*y^2/(x^2 + y^2)^3 - 12*x^2/(x^2 + y^2)^2 + 2/(x^2 + y^2)
show (f.diff(x,x)+f.diff(y,y)) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left( x, y \right) \ {\mapsto} \ \frac{8 \, x^{4}}{{\left(x^{2} + y^{2}\right)}^{3}} + \frac{8 \, x^{2} y^{2}}{{\left(x^{2} + y^{2}\right)}^{3}} - \frac{12 \, x^{2}}{{\left(x^{2} + y^{2}\right)}^{2}} + \frac{2}{x^{2} + y^{2}}
\newcommand{\Bold}[1]{\mathbf{#1}}\left( x, y \right) \ {\mapsto} \ \frac{8 \, x^{4}}{{\left(x^{2} + y^{2}\right)}^{3}} + \frac{8 \, x^{2} y^{2}}{{\left(x^{2} + y^{2}\right)}^{3}} - \frac{12 \, x^{2}}{{\left(x^{2} + y^{2}\right)}^{2}} + \frac{2}{x^{2} + y^{2}}
 
       
x, y= var ('x,y')