I have a few questions about the generalizations of concepts like integration and differentiation of single-valued functions of a single variable to vector-valued functions of several variables. All in the context of real analysis.
Beginning with scalar-valued functions of several variables (i.e. functions of the form ##f:\mathbb{R}^n\rightarrow\mathbb{R}##), my first question is regarding partial differentiation, definite and indefinite integration of this functions with respect to one variable. Am I right in assuming that in this case there are no differences between these concepts and the ones defined in single variable calculus (other than notation)? If I was programming a computer to perform these "operations" I wouldn't really have to teach it anything new, right? Just regard the function as a scalar-valued function of a single variable and treat the other "letters" you find as you would a constant like ##3## or ##5##.
The actual abstractions of differentiation, definite and indefinite integration of single-valued functions of a single variable to vector-valued functions of several variables have to do with Jacobian matrices and some abstraction of multiple integrals, right?
So the question is: if for functions of the form ##f:\mathbb{R}^n\rightarrow\mathbb{R}^m## derivatives abstract to Jacobian matrices, what do integrals abstract to?
I guess that one intermediary abstraction of (at least) definite integration for scalar-valued functions of several variables (##f:\mathbb{R}^n\rightarrow\mathbb{R}##) is the multiple integral. What is the abstraction of indefinite integrals in this case? What about further abstractions to functions of the form ##f:\mathbb{R}^n\rightarrow\mathbb{R}^m##?
Beginning with scalar-valued functions of several variables (i.e. functions of the form ##f:\mathbb{R}^n\rightarrow\mathbb{R}##), my first question is regarding partial differentiation, definite and indefinite integration of this functions with respect to one variable. Am I right in assuming that in this case there are no differences between these concepts and the ones defined in single variable calculus (other than notation)? If I was programming a computer to perform these "operations" I wouldn't really have to teach it anything new, right? Just regard the function as a scalar-valued function of a single variable and treat the other "letters" you find as you would a constant like ##3## or ##5##.
The actual abstractions of differentiation, definite and indefinite integration of single-valued functions of a single variable to vector-valued functions of several variables have to do with Jacobian matrices and some abstraction of multiple integrals, right?
So the question is: if for functions of the form ##f:\mathbb{R}^n\rightarrow\mathbb{R}^m## derivatives abstract to Jacobian matrices, what do integrals abstract to?
I guess that one intermediary abstraction of (at least) definite integration for scalar-valued functions of several variables (##f:\mathbb{R}^n\rightarrow\mathbb{R}##) is the multiple integral. What is the abstraction of indefinite integrals in this case? What about further abstractions to functions of the form ##f:\mathbb{R}^n\rightarrow\mathbb{R}^m##?
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