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1 Preliminaries
Minkowski (Hermann, 1864.6.22–1909.1.12) Russia-German mathematician and
physicist. Born in Alexotas, Lithuania, died in Göttingen. Successively studied at
the University of Berlin and University of Könisberg, became close friend with
Hilbert, received a doctor degree from the University of Göttingen in 1885. Became
a professor in 1892, served as a professor at the University of Könisberg in 1894,
a professor at the University of Zürich in Switzerland in 1896, a professor at the
University of Göttingen in 1902. The research range involved many aspects, had great
contributions in number theory, theory of convex body and mathematical physics etc.
The most famous contribution was to introduce the Minkowski four-dimensional
space-time in terms of mathematical physics, provided the mathematical foundation
for Einstein’s special theory of relativity. During studying in 1883 won a grand
prize of the French Academy of Sciences for the study of quadratic form. The
works had Geometrie der Zahlen (1896, 1910), Diophantische Approximationen:
Eine Einführung in die Zahlentheorie (Diophantus of Alexandria, about 210–290 or
about 246–330) (1907, 1961) and Raum und Zeit (1908) etc.
Einstein (Albert, 1879.3.14–1955.4.18) German–Swiss–American physicist. Born
in Ulm, Germany, died in Princeton, America. Received a Ph.D. from the University of Zürich in 1905. had served as the Patent Office in Bern. A professor of the
University of Zürich, Prague Deutsche University and Zürich Federal Institute of
Technology. Served as a professor at the University of Berlin 1913. Was appointed
director of the Kaiser William Institute for Physics in 1914 and a professor at the
Humboldt University of Berlin. Became a member of the Prussian Academy of
Sciences in 1913, became a member of the Royal Society in 1921. Won the Nobel
Prize in physics in 1921. Moved to the United States in 1933, served as a professor of
the Institute for Advanced Study in Princeton. Established the special theory of relativity in 1905, generalized into the general theory of relativity in 1916. Proposed the
concept of quantum of light, and explained the photoelectric effect, radiation process
and solid specific heat with the quantum theory. Had achievements in explaining the
Brownian movement and developed the quantum statistics. Later dedicated himself
to the establishment of a unified field theory of relativity. The works had Zur Elektrodynamik Bewegter Körpe (1905), Die Grundlage der Allgemeinen Relativitätstheorie (1916), The Evolution of Physics: The Growth of Ideas from early Concepts
to Relativity and Quanta (1938) and Unified Field Theory (1950) etc.
Problems 1
1.1 Prove = ϕϕψ + 2∇ϕ · ∇ψ + ψψϕ.
1.2 Prove ∇ · ( pa + q b) = p∇ · a + q∇ · b, where, both p and q are constants.
1.3 Verify ∇∇ ·(ϕa) = ∇ϕ∇ · a +ϕ∇∇ · a +∇ϕ ×∇ × a +(a ·∇)∇ϕ +(∇ϕ ·∇)a.
1.4 Verify
1 Preliminaries
Minkowski (Hermann, 1864.6.22–1909.1.12) Russia-German mathematician and
physicist. Born in Alexotas, Lithuania, died in Göttingen. Successively studied at
the University of Berlin and University of Könisberg, became close friend with
Hilbert, received a doctor degree from the University of Göttingen in 1885. Became
a professor in 1892, served as a professor at the University of Könisberg in 1894,
a professor at the University of Zürich in Switzerland in 1896, a professor at the
University of Göttingen in 1902. The research range involved many aspects, had great
contributions in number theory, theory of convex body and mathematical physics etc.
The most famous contribution was to introduce the Minkowski four-dimensional
space-time in terms of mathematical physics, provided the mathematical foundation
for Einstein’s special theory of relativity. During studying in 1883 won a grand
prize of the French Academy of Sciences for the study of quadratic form. The
works had Geometrie der Zahlen (1896, 1910), Diophantische Approximationen:
Eine Einführung in die Zahlentheorie (Diophantus of Alexandria, about 210–290 or
about 246–330) (1907, 1961) and Raum und Zeit (1908) etc.
Einstein (Albert, 1879.3.14–1955.4.18) German–Swiss–American physicist. Born
in Ulm, Germany, died in Princeton, America. Received a Ph.D. from the University of Zürich in 1905. had served as the Patent Office in Bern. A professor of the
University of Zürich, Prague Deutsche University and Zürich Federal Institute of
Technology. Served as a professor at the University of Berlin 1913. Was appointed
director of the Kaiser William Institute for Physics in 1914 and a professor at the
Humboldt University of Berlin. Became a member of the Prussian Academy of
Sciences in 1913, became a member of the Royal Society in 1921. Won the Nobel
Prize in physics in 1921. Moved to the United States in 1933, served as a professor of
the Institute for Advanced Study in Princeton. Established the special theory of relativity in 1905, generalized into the general theory of relativity in 1916. Proposed the
concept of quantum of light, and explained the photoelectric effect, radiation process
and solid specific heat with the quantum theory. Had achievements in explaining the
Brownian movement and developed the quantum statistics. Later dedicated himself
to the establishment of a unified field theory of relativity. The works had Zur Elektrodynamik Bewegter Körpe (1905), Die Grundlage der Allgemeinen Relativitätstheorie (1916), The Evolution of Physics: The Growth of Ideas from early Concepts
to Relativity and Quanta (1938) and Unified Field Theory (1950) etc.
Problems 1
1.1 Prove = ϕϕψ + 2∇ϕ · ∇ψ + ψψϕ.
1.2 Prove ∇ · ( pa + q b) = p∇ · a + q∇ · b, where, both p and q are constants.
1.3 Verify ∇∇ ·(ϕa) = ∇ϕ∇ · a +ϕ∇∇ · a +∇ϕ ×∇ × a +(a ·∇)∇ϕ +(∇ϕ ·∇)a.
1.4 Verify