Multivariable and vector calculus

इस पर आधारित

36 अनिवार्य पूर्वापेक्षाएँ (बिल्कुल नीचे तक)

आप सीखेंगे

• Compute partial and directional derivatives, gradients and Jacobian matrices.
• Solve constrained optimisation problems with Lagrange multipliers.
• Evaluate multiple, line and surface integrals and apply Green's theorem, Stokes's theorem and the divergence theorem.

परिभाषा

• The derivative of f : ℝⁿ → ℝᵐ at a is the linear map Df(a) such that f(a + h) = f(a) + Df(a)h + o(|h|) as h → 0.
• The gradient ∇f of a real-valued function collects its partial derivatives.

प्रमेय

• Chain rule: D(g ∘ f)(a) = Dg(f(a)) ∘ Df(a).
• Change of variables in multiple integrals uses the absolute value of the Jacobian determinant.
• Divergence theorem: the flux of F out of a bounded region V with smooth boundary equals ∭_V div F dV.
• If all partial derivatives of f exist and are continuous near a, then f is differentiable at a.

आम भ्रांतियाँ

• Thinking the existence of partial derivatives implies differentiability.
• Forgetting the Jacobian factor when changing variables.

आगे ले जाता है

वैकल्पिक खोजें

• Functors The chain rule says that taking derivatives respects composition; in categorical language, differentiation at a point is functorial.

अधिगम पथ

• Undergraduate mathematics major
• Applied mathematics

आगे पढ़ने के लिए

• John H. Hubbard and Barbara Burke Hubbard, Vector Calculus, Linear Algebra, and Differential Forms