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Calculus Volume 3

Key Equations

Calculus Volume 3Key Equations

Key Equations

Vertical trace f(a,y)=zf(a,y)=z for x=ax=a or f(x,b)=zf(x,b)=z for y=by=b
Level surface of a function of three variables f(x,y,z)=cf(x,y,z)=c
Partial derivative of ff with respect to xx fx=limh0f(x+h,y)f(x,y)hfx=limh0f(x+h,y)f(x,y)h
Partial derivative of ff with respect to yy fy=limk0f(x,y+k)f(x,y)kfy=limk0f(x,y+k)f(x,y)k
Tangent plane z=f(x0,y0)+fx(x0,y0)(xx0)+fy(x0,y0)(yy0)z=f(x0,y0)+fx(x0,y0)(xx0)+fy(x0,y0)(yy0)
Linear approximation L(x,y)=f(x0,y0)+fx(x0,y0)(xx0)+fy(x0,y0)(yy0)L(x,y)=f(x0,y0)+fx(x0,y0)(xx0)+fy(x0,y0)(yy0)
Total differential dz=fx(x0,y0)dx+fy(x0,y0)dy.dz=fx(x0,y0)dx+fy(x0,y0)dy.
Differentiability (two variables) f(x,y)=f(x0,y0)+fx(x0,y0)(xx0)+fy(x0,y0)(yy0)+E(x,y),f(x,y)=f(x0,y0)+fx(x0,y0)(xx0)+fy(x0,y0)(yy0)+E(x,y),
where the error term EE satisfies
lim(x,y)(x0,y0)E(x,y)(xx0)2+(yy0)2=0.lim(x,y)(x0,y0)E(x,y)(xx0)2+(yy0)2=0.
Differentiability (three variables) f(x,y)=f(x0,y0,z0)+fx(x0,y0,z0)(xx0)+fy(x0,y0,z0)(yy0)+fz(x0,y0,z0)(zz0)+E(x,y,z),f(x,y)=f(x0,y0,z0)+fx(x0,y0,z0)(xx0)+fy(x0,y0,z0)(yy0)+fz(x0,y0,z0)(zz0)+E(x,y,z),
where the error term EE satisfies
lim(x,y,z)(x0,y0,z0)E(x,y,z)(xx0)2+(yy0)2+(zz0)2=0.lim(x,y,z)(x0,y0,z0)E(x,y,z)(xx0)2+(yy0)2+(zz0)2=0.
Chain rule, one independent variable dzdt=zx·dxdt+zy·dydtdzdt=zx·dxdt+zy·dydt
Chain rule, two independent variables dzdu=zxxu+zyxudzdu=zxxu+zyxu
dzdv=zxxv+zyyvdzdv=zxxv+zyyv
Generalized chain rule wtj=wx1x1tj+wx2x1tj++wxmxmtjwtj=wx1x1tj+wx2x1tj++wxmxmtj
directional derivative (two dimensions) Duf(a,b)=limh0f(a+hcosθ,b+hsinθ)f(a,b)hDuf(a,b)=limh0f(a+hcosθ,b+hsinθ)f(a,b)h
or
Duf(x,y)=fx(x,y)cosθ+fy(x,y)sinθDuf(x,y)=fx(x,y)cosθ+fy(x,y)sinθ
gradient (two dimensions) f(x,y)=fx(x,y)i+fy(x,y)jf(x,y)=fx(x,y)i+fy(x,y)j
gradient (three dimensions) f(x,y,z)=fx(x,y,z)i+fy(x,y,z)j+fz(x,y,z)kf(x,y,z)=fx(x,y,z)i+fy(x,y,z)j+fz(x,y,z)k
directional derivative (three dimensions) Duf(x,y,z)=f(x,y,z)·u=fx(x,y,z)cosα+fy(x,y,z)cosβ+fx(x,y,z)cosγDuf(x,y,z)=f(x,y,z)·u=fx(x,y,z)cosα+fy(x,y,z)cosβ+fx(x,y,z)cosγ
Discriminant D=fxx(x0,y0)fyy(x0,y0)(fxy(x0,y0))2D=fxx(x0,y0)fyy(x0,y0)(fxy(x0,y0))2
Method of Lagrange multipliers, one constraint f(x0,y0)=λg(x0,y0)g(x0,y0)=0f(x0,y0)=λg(x0,y0)g(x0,y0)=0
Method of Lagrange multipliers, two constraints f(x0,y0,z0)=λ1g(x0,y0,z0)+λ2h(x0,y0,z0)g(x0,y0,z0)=0h(x0,y0,z0)=0f(x0,y0,z0)=λ1g(x0,y0,z0)+λ2h(x0,y0,z0)g(x0,y0,z0)=0h(x0,y0,z0)=0
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