B2_Partial_Derivatives_2a_Blank - MATH10121 B2 B2.1...

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B2.1MATH10121 B2functionsinmorethanonedimensionEquations, such asxyza “saddle”z= 1 +x2y2describe a surfacein space, whilealso, defininga functionz=f(x, y)wherefmaps(x, y)to1 +x2y2with domainR2andrangeR(a ‘real-valued function of two independent variables’)More examples:excosymapsR2R1￿1 +x2+y2+z2mapsR3Rsin(rt)mapsR2Rtcost￿i+tsint￿jmapsRR2￿k×￿rmapsR3R3(the last two are called ‘vector-valued functions’)B2.2functions fromRntoRmFunctions may have domains inRnand ranges inRmA real-valued function has its range inRA vector-valued function has range inRmform >1A vector-valued function with range inRmcan be thought of asmreal-valued functions(each representing one coordinate of a vector)￿f1(x1, x2, . . . xn), f2(x1, . . . xn), . . . fm(x1, . . . xn)￿A function of (say) two variables is often written asf(x, y)orf(·,·)where the order of the variables is important, that isf(x, y)￿=f(y, x)Example:ifg(x, y) =excosywhat isg(y, x)?Functions of any number of variables can arise, e.g.S(t, x, y, z, P, T)mappingR6RAs for functions of a single-variable:a function isdefined for all elementsin its domain.a function maps any one element in its domaintoonly one elementin its range.
B2.3‘partial’ derivativesA function of two variables, such asf(x, y),has two firstderivatives:(a)a derivative with respect tox(b)a derivative with respect toyA function ofnvariables hasnfirst derivatives.Together, these derivatives make up the fullfirst-order differentiation of the function.Each one, separately, is called a ‘partial derivative’.Notation:the partial derivative off(x, y)withrespect toxcan be written as eitherfxorfxorxforxfNote.is called ‘del’ or ‘partial’do not writedforDefinition:the partial derivativefxis defined asfx(x, y) = limh0f(x+h, y)f(x, y)hin whichyis simply treated as a constant.More Notation:sometimes we might writefx￿￿yorfx￿￿￿yorx￿￿￿yforx￿￿yfto stress thatyis treated as a constant in calculatingthe partial derivative with respect toxB2.4more on partial derivativesPartial derivatives are found in exactly the same way asordinary derivativesExample 1.φ(x, y) =x2+y22xyfindφxandφyExample 2.v(t, r) = sin(ctr)findvtandvrExample 3.g(t, x, y) =ex23xytfindgt,gxandgy
B2.5chain-rule for partial derivativesThe chain-rule extends to more than one variable.for 1 variable:ddxf￿u(x)￿=f￿￿u(x)￿u￿(x) =dfdududxfor 2 variables:suppose we havef(u, v)and thatu=u(x, y)andv=v(x, y)that is, we havef￿u(x, y), v(x, y)￿,thenx￿￿yy￿￿x￿fx=fuux+fvvxfy=fuuy+fvvy￿u￿￿vv￿￿uwhich changes variables from(u, v)to(x, y)on left:fx￿￿y&fy￿￿xand on right:fu￿￿v&fv￿￿uExample:z=xywithx=rcosθ,y=rsinθ,find the partial derivativeszrandzθB2.6more on chain-rule

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