Become a Multivariable Calculus Pro!

Write formulas for the isometries in terms of a complex variable $z=x+iy.$

Let f1,…,fr be complex polynomials in the variables x1,…,xn let V be the variety of their common zeros, and let I be the ideal of the polynomial ring $R=C[x1,\dots ,xn]$ that they generate. Define a homomorphism from the quotient ring $\bar{R}=Rx2F$; I to the ring RR of continuous, complex-valued functions on V.

Write formulas for the indicated partial derivatives for the multivariable function.
${\displaystyle f(x,y)=7x^2+9xy+4y^3}$
a)${\displaystyle \frac{\partial f}{\partial x}}$
b) $\frac{\partial f}{\partial y}$
c)${\displaystyle \frac{\partial f}{\partial x}{\mid }_{y=9}}$

Use Green's Theorem to evaluate the line integral

${\int }_C(y+e^x)dx+(6x+\mathrm{cosy})dy$

where $C$ is triangle with vertices $(0,0), (0,2)$ and $(2,2)$ oriented counterclockwise.

Use Green's Theorem in the form of this equation to prove Green's first identity, where D and C satisfy the hypothesis of Green's Theorem and the appropriate partial derivatives of f and g exist and are continuous. (The quantity grad $g\times n=Dng$ occurs in the line integral. This is the directional derivative in the direction of the normal vector n and is called the normal derivative of g.)
${\displaystyle {\oint }_{c}F\cdot nds=\int {\int }_D÷F(x,y)dA}$

Write first and second partial derivatives
${\displaystyle g(r,t)=t\ln r+11r{t}^7-5\left(8^r\right)-tr}$
a)${\displaystyle g_r}$
b)${\displaystyle g_{rr}}$
c)${\displaystyle g_{rt}}$
d)${\displaystyle g_t}$
e)${\displaystyle g_t}$