MediMath Science

This blog presents undergraduate-level vector calculus with examples and perspectives drawn from artificial intelligence. It was created in response to the limitations of many existing calculus textbooks in addressing the conceptual demands of this rapidly evolving field. Before we begin, it is important to clarify how the variable \(\mathbf{x}\) will be used throughout the text. In classical calculus, the variable \(\mathbf{x}\)  typically denotes a physical coordinate, such as position in space. In deep learning, however, \(\mathbf{x}\) often represents a token or a feature vector that encodes complex structures, including language, images, or speech. This shift reflects a transition from modeling physical space to modeling abstract, high-dimensional data spaces. 1. Vectors Vectors  are mathematical entities characterized by both magnitude and direction. They are essential for describing physical quantities such as blood flow velocity, forces acting on joints, an...

In today’s financial markets, a troubling asymmetry has emerged between those who sell technological dreams and those who buy them. Founders, CEOs, and early investors often understand how distant true commercialization remains, while many young retail investors—driven by optimism and headlines—see only the promise, not the timeline. This imbalance creates a structural divide: the informed side monetizes expectations, while the uninformed side absorbs the losses. Quantum computing offers a striking example. The 2025 Nobel Prize in Physics was awarded to John Clarke, Michel H. Devoret, and John M. Martinis “for the discovery of macroscopic quantum mechanical tunnelling and energy quantisation in an electric circuit.” Their work revealed that quantum effects—once confined to the microscopic world—can emerge in circuits large enough to see, laying the foundation for today’s superconducting qubits. Quantum systems promise radically new ways of computation through superposition, entangleme...