设fx(x0,y0)?fy(x0,y0)?0,令:fxx(x0,y0)?A, fxy(x0,y0)?B, fyy(x0,y0)?C??A?0,(x0,y0)为极大值2AC?B?0时,???A?0,(x0,y0)为极小值??2则:值?AC?B?0时, 无极?AC?B2?0时, 不确定???重积分及其应用:
??f(x,y)dxdy???f(rcos?,rsin?)rdrd?DD?曲面z?f(x,y)的面积A???D??z???z?1???????y??dxdy?x????22M平面薄片的重心:x?x?M??x?(x,y)d?D???(x,y)d?DD, y?MyM???y?(x,y)d?D???(x,y)d?DD
平面薄片的转动惯量:对于x轴Ix???y2?(x,y)d?, 对于y轴Iy???x2?(x,y)d?平面薄片(位于xoy平面)对z轴上质点M(0,0,a),(a?0)的引力:F?{Fx,Fy,Fz},其中:Fx?f??D?(x,y)xd?(x2?y2?a2)2, Fy?f??3D?(x,y)yd?(x2?y2?a2)2, Fz??fa??3D?(x,y)xd?(x2?y2?a)322柱面坐标和球面坐标:
?x?rcos??柱面坐标:f(x,y,z)dxdydz????F(r,?,z)rdrd?dz,?y?rsin?, ??????z?z?其中:F(r,?,z)?f(rcos?,rsin?,z)?x?rsin?cos??2球面坐标:?y?rsin?sin?, dv?rd??rsin??d??dr?rsin?drd?d??z?rcos??2?
?r(?,?)???f(x,y,z)dxdydz????F(r,?,?)r??2sin?drd?d???d??d?00?F(r,?,?)r02sin?dr重心:x?1M???x?dv, y???1M???y?dv, z???1M???z?dv, 其中M?x?????dv???转动惯量:Ix????(y2?z2)?dv, Iy????(x2?z2)?dv, Iz????(x2?y2)?dv曲线积分:
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第一类曲线积分(对弧长的曲线积分):?x??(t)设f(x,y)在L上连续,L的参数方程为:, (??t??),则:??y??(t)?L?x?t22??f(x,y)ds??f[?(t),?(t)]?(t)??(t)dt (???) 特殊情况:??y??(t)??第二类曲线积分(对坐标的曲线积分):?x??(t)设L的参数方程为,则:?y??(t)???P(x,y)dx?Q(x,y)dy???{P[?(t),?(t)]??(t)?Q[?(t),?(t)]??(t)}dtL两类曲线积分之间的关系:?Pdx?Qdy??(Pcos??Qcos?)ds,其中?和?分别为LLL上积分起止点处切向量的方向角。?Q?P?Q?P格林公式:(?)dxdy?Pdx?Qdy格林公式:(?)dxdy??Pdx?Qdy??????x?y?x?yDLDL?Q?P1当P??y,Q?x,即:??2时,得到D的面积:A???dxdy??xdy?ydx?x?y2LD·平面上曲线积分与路径无关的条件:1、G是一个单连通区域;2、P(x,y),Q(x,y)在G内具有一阶连续偏导数,且减去对此奇点的积分,注意方向相反!·二元函数的全微分求积:?Q?P在=时,Pdx?Qdy才是二元函数u(x,y)的全微分,其中:?x?y(x,y)?Q?P=。注意奇点,如(0,0),应?x?y
u(x,y)?(x0,y0)?P(x,y)dx?Q(x,y)dy,通常设x0?y0?0。曲面积分:
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22对面积的曲面积分:f(x,y,z)ds?f[x,y,z(x,y)]1?z(x,y)?z(x,y)dxdyxy?????Dxy对坐标的曲面积分:??P(x,y,z)dydz?Q(x,y,z)dzdx?R(x,y,z)dxdy,其中:?号;??R(x,y,z)dxdy????R[x,y,z(x,y)]dxdy,取曲面的上侧时取正?Dxy号;??P(x,y,z)dydz????P[x(y,z),y,z]dydz,取曲面的前侧时取正?Dyz
??Q(x,y,z)dzdx????Q[x,y(z,x),z]dzdx,取曲面的右侧时取正号。?Dzx两类曲面积分之间的关系:??Pdydz?Qdzdx?Rdxdy???(Pcos??Qcos??Rcos?)ds??高斯公式:
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???(??P?Q?R??)dv???Pdydz?Qdzdx?Rdxdy???(Pcos??Qcos??Rcos?)ds?x?y?z??高斯公式的物理意义——通量与散度:??P?Q?R?散度:div????,即:单位体积内所产生的流体质量,若div??0,则为消失...?x?y?z??通量:??A?nds???Ands???(Pcos??Qcos??Rcos?)ds,?因此,高斯公式又可写成:???divAdv???Ands?????斯托克斯公式——曲线积分与曲面积分的关系:
??(??R?Q?P?R?Q?P?)dydz?(?)dzdx?(?)dxdy??Pdx?Qdy?Rdz?y?z?z?x?x?y?cos???yQcos???zR
dydzdzdxdxdycos?????上式左端又可写成:??????x?y?z?x??PQRP?R?Q?P?R?Q?P空间曲线积分与路径无关的条件:?, ?, ??y?z?z?x?x?yijk????旋度:rotA??x?y?zPQR???向量场A沿有向闭曲线?的环流量:?Pdx?Qdy?Rdz??A?tds??常数项级数:
1?qn等比数列:1?q?q???q?1?q(n?1)n 等差数列:1?2?3???n?2111调和级数:1?????是发散的23n2n?1级数审敛法:
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1、正项级数的审敛法——根植审敛法(柯西判别法):???1时,级数收敛?设:??limnun,则???1时,级数发散n?????1时,不确定?2、比值审敛法:???1时,级数收敛U?设:??limn?1,则???1时,级数发散n??Un???1时,不确定?3、定义法:sn?u1?u2???un;limsn存在,则收敛;否则发散。n??
交错级数u1?u2?u3?u4??(或?u1?u2?u3??,un?0)的审敛法——莱布尼兹定理:? ?un?un?1如果交错级数满足s?u1,其余项rn的绝对值rn?un?1。?limu?0,那么级数收敛且其和??n??n绝对收敛与条件收敛:
(1)u1?u2???un??,其中un为任意实数;(2)u1?u2?u3???un??如果(2)收敛,则(1)肯定收敛,且称为绝对收敛级数;如果(2)发散,而(1)收敛,则称(1)为条件收敛级数。 1(?1)n调和级数:?n发散,而?n收敛;1 级数:?n2收敛;p?1时发散1 p级数:?np p?1时收敛幂级数:
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考研数学公式大全(考研必备)
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