……庞大而古老的领域，而且现在关于其中有多少内容还值得研究是有争议的。不过要注意，对于常微分方程组，可以用一些很巧妙的方式应用线性代数。关于面向应用的高等微积分，有 MIT 老牌的 Hildebrand 的书——他非常懂行，读起来也够容易，如果需要他讲到的某个主题，那是个很好的参考。最近几十年来，纯数学系想要……
…huge, old field, and there is some question about how much of that deserves study now. Do notice that for systems of ordinary differential equations, get to apply some linear algebra in cute ways. For advanced calculus for applications, there is the old MIT Hildebrand -- he knows what he is talking about, is easy enough to read, and a good place to go if need one of his topics. In recent decades, the pure math departments wanted to…
……书，Coddington 和 Levinson 的书则高深得多）——现在搞基础交流电（AC）电路理论简直跟吃冰淇淋一样轻松。找一本或几本更传统的书学高等微积分，比如 MIT 的老牌心头好 Hildebrand 的《应用高等微积分》（Advanced Calculus for Applications）、Fleming 的《多元函数论》（Functions of Several Variables）、Buck 的《高等微积分》（Advanced Calculus）——现在看麦克斯韦方程组至少能看懂数学部分了。而且还能用梯度在最大……做最速下降法（steepest descent）……
…book, Coddington and Levinson is much more advanced) -- now can do basic AC circuit theory like eating ice cream. Advanced calculus from one or several more traditional books, e.g., the old MIT favorite Hildebrand, Advanced Calculus for Applications, Fleming, Functions of Several Variables, Buck, Advanced Calculus -- can now look at Maxwell's equations and understand at least the math. And can work with the gradient for steepest descent in the maximum…
……不再真正“现代”了！去买本二手书吧；我当时就是这么干的。可别买新版！不过闲话少说，不妨快速过一遍 MIT 的老标准教材：Francis B. Hildebrand 的《应用高等微积分》（Advanced Calculus for Applications，Prentice-Hall，恩格尔伍德克利夫斯，新泽西州，1962年）。里面有很多有趣的东西，但证明实在太太太少了。所以，我建议先略过它，等学了足够的实分析（real analysis），能自己猜出或理解证明时再回来看……
…longer really "modern"! Get a used copy; that's what I did. You don't want a newer edition! But with that out of the way, might take a fast pass through an old MIT standard Francis B.\ Hildebrand, {\it Advanced Calculus for Applications,\/} Prentice-Hall, Englewood Cliffs, NJ, 1962.\ \ There's a lot of fun stuff in there, but it's TOO short on proofs. So, I'd pass it up and return later when know enough real analysis to guess or understand…
……卡拉西奥多里（Carathéodory）的结果，但一般来说啃完那些得少睡很多觉！我上一门由一位 MIT 博士教的课，用的教材显然长期是 MIT 的标准教材——Francis B. Hildebrand 的《应用高等微积分》（Advanced Calculus for Applications，Prentice-Hall，恩格尔伍德克利夫斯，新泽西州，1962年）。所以，是的，你能学到通过无穷级数求解以及边值问题（boundary value problems）。这本书极度缺乏证明，而要理解这种材料……
…result of Caratheodory, but in general could lose a lot of sleep working through that! I had a course from a Ph.D. from MIT from the book, apparently long a standard at MIT, Francis B.\ Hildebrand, {\it Advanced Calculus for Applications,\/} Prentice-Hall, Englewood Cliffs, NJ, 1962.\ \ So, yes, can find out about solutions via infinite series and boundary value problems. The book was very short on proofs, and to take such material…
……避开了 Buck、Fleming、Spivak，当然还有 Henri Cartan 的《微分形式》（Differential Forms，现在有英文版了）。我甚至避开了长期在 MIT 使用的经典应用高等微积分教材——Francis B. Hildebrand 的《应用高等微积分》（Advanced Calculus for Applications）。至于 Loomis 和 Sternberg，我赞同你的看法，纸质版和 PDF 我都有。既然我都提到了这类高等微积分，那我想尽力挽救广大学生：同学们，这里有个秘密。那……
…avoided Buck, Fleming, Spivak, and of course also, now in English, Henri Cartan, 'Differential Forms'. I even avoided the classic applied advanced calculus text, long used at MIT, Francis B. Hildebrand, 'Advanced Calculus for Applications'. For Loomis and Sternberg, I agree with you, and have both the hard copy and the PDF. Since I've mentioned such advanced calculus, I will try to save many students: Students, there's a secret. The…
……来表述这两者。要把宏大的主题写得既清晰、简练又准确，往往并不容易！我接触到斯图姆-刘维尔理论（Sturm-Liouville theory）是在一门课上，教材是相当知名的 MIT Hildebrand 著《应用高等微积分》（Advanced Calculus for Applications），授课老师刚拿到 MIT 博士学位不久。顺便说一句，那本书网上有免费的 PDF。对我来说，那个课题看着就像振动弦之类的数学，但那门课……
…to formulate those two. Writing clearly, briefly, and precisely about big subjects is not always easy! The time I encountered Strum-Louville theory was a course from the fairly well known Hildebrand, Advanced Calculus for Applications from MIT and by a professor with a recent MIT Ph.D. By the way, that book is available in PDF on the Internet for free. To me the topic looked like the math of vibrating strings, etc., but the course…
……庞大而古老的领域，而且现在关于其中有多少内容还值得研究是有争议的。不过要注意，对于常微分方程组，可以用一些很巧妙的方式应用线性代数。关于面向应用的高等微积分，有 MIT 老牌的 Hildebrand 的书——他非常懂行，读起来也够容易，如果需要他讲到的某个主题，那是个很好的参考。最近几十年来，纯数学系想要……
…huge, old field, and there is some question about how much of that deserves study now. Do notice that for systems of ordinary differential equations, get to apply some linear algebra in cute ways. For advanced calculus for applications, there is the old MIT Hildebrand -- he knows what he is talking about, is easy enough to read, and a good place to go if need one of his topics. In recent decades, the pure math departments wanted to…
……书，Coddington 和 Levinson 的书则高深得多）——现在搞基础交流电（AC）电路理论简直跟吃冰淇淋一样轻松。找一本或几本更传统的书学高等微积分，比如 MIT 的老牌心头好 Hildebrand 的《应用高等微积分》（Advanced Calculus for Applications）、Fleming 的《多元函数论》（Functions of Several Variables）、Buck 的《高等微积分》（Advanced Calculus）——现在看麦克斯韦方程组至少能看懂数学部分了。而且还能用梯度在最大……做最速下降法（steepest descent）……
…book, Coddington and Levinson is much more advanced) -- now can do basic AC circuit theory like eating ice cream. Advanced calculus from one or several more traditional books, e.g., the old MIT favorite Hildebrand, Advanced Calculus for Applications, Fleming, Functions of Several Variables, Buck, Advanced Calculus -- can now look at Maxwell's equations and understand at least the math. And can work with the gradient for steepest descent in the maximum…
……不再真正“现代”了！去买本二手书吧；我当时就是这么干的。可别买新版！不过闲话少说，不妨快速过一遍 MIT 的老标准教材：Francis B. Hildebrand 的《应用高等微积分》（Advanced Calculus for Applications，Prentice-Hall，恩格尔伍德克利夫斯，新泽西州，1962年）。里面有很多有趣的东西，但证明实在太太太少了。所以，我建议先略过它，等学了足够的实分析（real analysis），能自己猜出或理解证明时再回来看……
…longer really "modern"! Get a used copy; that's what I did. You don't want a newer edition! But with that out of the way, might take a fast pass through an old MIT standard Francis B.\ Hildebrand, {\it Advanced Calculus for Applications,\/} Prentice-Hall, Englewood Cliffs, NJ, 1962.\ \ There's a lot of fun stuff in there, but it's TOO short on proofs. So, I'd pass it up and return later when know enough real analysis to guess or understand…
……卡拉西奥多里（Carathéodory）的结果，但一般来说啃完那些得少睡很多觉！我上一门由一位 MIT 博士教的课，用的教材显然长期是 MIT 的标准教材——Francis B. Hildebrand 的《应用高等微积分》（Advanced Calculus for Applications，Prentice-Hall，恩格尔伍德克利夫斯，新泽西州，1962年）。所以，是的，你能学到通过无穷级数求解以及边值问题（boundary value problems）。这本书极度缺乏证明，而要理解这种材料……
…result of Caratheodory, but in general could lose a lot of sleep working through that! I had a course from a Ph.D. from MIT from the book, apparently long a standard at MIT, Francis B.\ Hildebrand, {\it Advanced Calculus for Applications,\/} Prentice-Hall, Englewood Cliffs, NJ, 1962.\ \ So, yes, can find out about solutions via infinite series and boundary value problems. The book was very short on proofs, and to take such material…
……避开了 Buck、Fleming、Spivak，当然还有 Henri Cartan 的《微分形式》（Differential Forms，现在有英文版了）。我甚至避开了长期在 MIT 使用的经典应用高等微积分教材——Francis B. Hildebrand 的《应用高等微积分》（Advanced Calculus for Applications）。至于 Loomis 和 Sternberg，我赞同你的看法，纸质版和 PDF 我都有。既然我都提到了这类高等微积分，那我想尽力挽救广大学生：同学们，这里有个秘密。那……
…avoided Buck, Fleming, Spivak, and of course also, now in English, Henri Cartan, 'Differential Forms'. I even avoided the classic applied advanced calculus text, long used at MIT, Francis B. Hildebrand, 'Advanced Calculus for Applications'. For Loomis and Sternberg, I agree with you, and have both the hard copy and the PDF. Since I've mentioned such advanced calculus, I will try to save many students: Students, there's a secret. The…
……来表述这两者。要把宏大的主题写得既清晰、简练又准确，往往并不容易！我接触到斯图姆-刘维尔理论（Sturm-Liouville theory）是在一门课上，教材是相当知名的 MIT Hildebrand 著《应用高等微积分》（Advanced Calculus for Applications），授课老师刚拿到 MIT 博士学位不久。顺便说一句，那本书网上有免费的 PDF。对我来说，那个课题看着就像振动弦之类的数学，但那门课……
…to formulate those two. Writing clearly, briefly, and precisely about big subjects is not always easy! The time I encountered Strum-Louville theory was a course from the fairly well known Hildebrand, Advanced Calculus for Applications from MIT and by a professor with a recent MIT Ph.D. By the way, that book is available in PDF on the Internet for free. To me the topic looked like the math of vibrating strings, etc., but the course…