作业帮微积分 求全微分 微积分 求全微分
来源：学生作业帮 编辑：作业帮 分类：数学作业 时间：2024/07/15 05:59:47
微积分 求全微分
微积分 求全微分
微积分 求全微分
du=e^(xy)d(xy)+cos(yz)d(yz)
=e^(xy)(xdy+ydx)+cos(yz)(ydz+zdy)
=ye^(xy)dx+[xe^(xy)+zcos(yz)]dy+zcos(yz)dz
下面由cos(^2)x+cos(^2)y+cos(^2)z=1求dz.
对上式两边微分得：-2cosxsinxdx-2cosysinydy-2coszsinzdz=0
即 sin(2x)dx+sin(2y)dy+sin(2z)dz=0
故：dz=-[sin(2x)/sin(2z)]dx-[sin(2y)/sin(2z)]dy
将上式代入第三行得：
du=ye^(xy)dx+[xe^(xy)+zcos(yz)]dy+zcos(yz){-[sin(2x)/sin(2z)]dx-[sin(2y)/sin(2z)]dy}
={ye^(xy)-zcos(yz))[sin(2x)/sin(2z)]}dx+{xe^(xy)+zcos(yz)-zcos(yz)[sin(2y)/sin(2z)]}dy
注：课后的答案是错误的!
再问： 应该是ycos(xy)dz不是zcos(xy)dz
再问： 课后答案是什么？
再答： 1、是的, 应该是ycos(xy)dz，谢谢你。修改如下(和前面运算符号又空格的地方(共有4处),都是修改过的):
du=e^(xy)d(xy)+cos(yz)d(yz)
=e^(xy)(xdy+ydx)+cos(yz)(ydz+zdy)
=ye^(xy)dx+[xe^(xy)+zcos(yz)]dy+ ycos(yz)dz
下面由cos(^2)x+cos(^2)y+cos(^2)z=1求dz。
对上式两边微分得：-2cosxsinxdx-2cosysinydy-2coszsinzdz=0
即 sin(2x)dx+sin(2y)dy+sin(2z)dz=0
故：dz=-[sin(2x)/sin(2z)]dx-[sin(2y)/sin(2z)]dy
将上式代入第三行得：
du=ye^(xy)dx+[xe^(xy)+zcos(yz)]dy + ycos(yz){-[sin(2x)/sin(2z)]dx-[sin(2y)/sin(2z)]dy}
={ye^(xy)- ycos(yz))[sin(2x)/sin(2z)]}dx+{xe^(xy)+zcos(yz)- ycos(yz)[sin(2y)/sin(2z)]}dy
2、这是一课本上的习题，它给的答案是错误的。
=e^(xy)(xdy+ydx)+cos(yz)(ydz+zdy)
=ye^(xy)dx+[xe^(xy)+zcos(yz)]dy+zcos(yz)dz
下面由cos(^2)x+cos(^2)y+cos(^2)z=1求dz.
对上式两边微分得：-2cosxsinxdx-2cosysinydy-2coszsinzdz=0
即 sin(2x)dx+sin(2y)dy+sin(2z)dz=0
故：dz=-[sin(2x)/sin(2z)]dx-[sin(2y)/sin(2z)]dy
将上式代入第三行得：
du=ye^(xy)dx+[xe^(xy)+zcos(yz)]dy+zcos(yz){-[sin(2x)/sin(2z)]dx-[sin(2y)/sin(2z)]dy}
={ye^(xy)-zcos(yz))[sin(2x)/sin(2z)]}dx+{xe^(xy)+zcos(yz)-zcos(yz)[sin(2y)/sin(2z)]}dy
注：课后的答案是错误的!
再问： 应该是ycos(xy)dz不是zcos(xy)dz
再问： 课后答案是什么？
再答： 1、是的, 应该是ycos(xy)dz，谢谢你。修改如下(和前面运算符号又空格的地方(共有4处),都是修改过的):
du=e^(xy)d(xy)+cos(yz)d(yz)
=e^(xy)(xdy+ydx)+cos(yz)(ydz+zdy)
=ye^(xy)dx+[xe^(xy)+zcos(yz)]dy+ ycos(yz)dz
下面由cos(^2)x+cos(^2)y+cos(^2)z=1求dz。
对上式两边微分得：-2cosxsinxdx-2cosysinydy-2coszsinzdz=0
即 sin(2x)dx+sin(2y)dy+sin(2z)dz=0
故：dz=-[sin(2x)/sin(2z)]dx-[sin(2y)/sin(2z)]dy
将上式代入第三行得：
du=ye^(xy)dx+[xe^(xy)+zcos(yz)]dy + ycos(yz){-[sin(2x)/sin(2z)]dx-[sin(2y)/sin(2z)]dy}
={ye^(xy)- ycos(yz))[sin(2x)/sin(2z)]}dx+{xe^(xy)+zcos(yz)- ycos(yz)[sin(2y)/sin(2z)]}dy
2、这是一课本上的习题，它给的答案是错误的。