Derivative of a Multivariable Function from Definition in Vector Spaces

samer88
Messages
7
Reaction score
0

Homework Statement


determine the derivative of f(x,y,z)=(x^2-2xy+z,y^2+z^2) directly from the definition where f:R^3------->R^2


Homework Equations





The Attempt at a Solution

 
Physics news on Phys.org
The derivative or derivatives? If it is the derivatives just take the partial derivatives inside the vector
 
no it is the derivative
 
And it does not say with respect to what?
I would guess taking all the first derivatives
f(x,y,z)=(x^2-2xy+z,y^2+z^2)
f'(x) = (2x-2y,0)
f'(y) = (-2x,2y)
f'(z) = (1,2z)
But I guess the question is to advanced for me:-(
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
Back
Top